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Logic

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- Critical thinking skill.

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Open and Closed Sentences

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- Open: Sentence with variable. ex:) x+2=8, It is my favorite color. - Closed: Sentence without a variable. ex:) 6+2=8, The grass is green.

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Open Sentences

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- Open: Sentence with variable. ex:) x+2=8, It is my favorite color.

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Closed Sentences

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- Closed: Sentence without a variable. ex:) 6+2=8, The grass is green.

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Sentences have a truth value

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- They are either true, false, or open. - They must be closed to be truth/false. - ex:) True: A square has equal side lengths. False: A pentagon has 67 sides. Open: It has a right angle.

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Negation

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- Giving the opposite truth value. - Symbol ~ Symbols: Representations of sentences, usually in letter form. ex.) P: I passed my test. negation: ~P I did not pass my test.

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And / Or

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- And (Conjunction): Two things depend on each other. Symbol: ^. To be true, both parts must be true. - Or (Disjunction): Two things do not depend on each other. Symbol: V. - Examples: And- C= All cats bark. E= 7+4=11 C^E (False) ~C^E (True) C^~E (False) ~E^~C (False) Or- M= I go to the mall. B: I buy sneakers. MVB (True) ~MVB (True) MV~B (True) ~MV~B (False)

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And (Conjunction)

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wo things depend on each other. Symbol: ^. To be true, both parts must be true.

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Or (Disjunction)

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Two things do not depend on each other. Symbol: V.

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Negation of Phrases

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- Distribute the negative. - Flip the sign. - Symbol: ~. - Ex.) ~(P^Q) : ~PV~Q ~(~CVB) : C^~B

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Conditional

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- Means to imply. - A compound sentence. - Every action has a reaction. - Parts of the sentence: *hypothesis- Sentence or phrase that starts with \"if.\" *conclusion- Sentence or phrase that starts with \"then.\" ex.) If I eat a lot of icecream, then I get a brainfreeze. - Symbol: -> ex.) P: I go on a field trip, W: I miss schoolwork. P->W (Truth Value- T->T equals T) P Q P-> Q T T T T F F F T T F F T

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hypothesis

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Sentence or phrase that starts with \"if.\"

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conclusion

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Sentence or phrase that starts with \"then.\" ex.) If I eat a lot of icecream, then I get a brainfreeze. - Symbol: ->

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T

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T T

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F

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T F

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T

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F T

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T

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F F

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Biconditional

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- If and only if. - Symbol: . ex.) P: A triangle is isosceles. (T) Q: A triangle has 2 congruent sides. (T) P Q (T) P Q PQ T T T T F F F T F F F T

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Inverse

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- Changing the truth value of the conditional. - Change the sign. - ex.) If I like chocolate, then I eat kisses. If I do not like chocolate, then I do not eat kisses.

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Converse

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- Changing the order of the conditional. - Change order. - ex.) If I put on shoes, then I tie my shoes. If I tie my shoes, then I put on my shoes.

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Contrapositive

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- Logically equilvilant. - Change the order and the sign. - Same truth value. - ex.) If I get married, then I say I do. If I do not say I do, then I do not get married.

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categorical proposition

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A proposition that relates two classes, or categories ex. American Idol contestants hope for recognition

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4 Types of Categorical Propositions

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(1) those that assert that the whole subject class is included in the predicate class, (2) those that assert that part of the subject class is included in the predicate class, (3) those that assert that the whole subject class is excluded from the predicate class, and (4) those that assert that part of the subject class is excluded from the predicate class.

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standard form categorical proposition

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A categorical proposition that expresses these relations with complete clarity

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Forms of Standard form categorical proposition

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All S are P. No S are P. Some S are P. Some S are not P.

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quantifiers

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they specify how much of the subject class is included in or excluded from the predicate class. (ex. all, no, or some)

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copula

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they link (or \"couple\") the subject term with the predicate term. (ex. are and are not)

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quality

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either affirmative or negative depending on whether it affirms or denies class membership

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affirmative quality

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ex: \"All S are P\" and \"Some S are P\"

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negative quality

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\"No S are P\" and \"Some S are not P\"

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quantity

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categorical proposition is either universal or particular, depending on whether the statement makes a claim about every member or just some member of the class denoted by the subject term

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universal statements

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ex: \"All S are P\" and \"No S are P\" each assert something about every member of the S class

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particular statements

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ex: \"Some S are P \" and \"Some S are not P \" assert something about one or more members of the S class

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A proposition

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universal affirmative

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E proposition

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universal negative

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I proposition

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particular affirmative

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O proposition

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particular negative

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distribution

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an attribute of the terms (subject and predicate) of propositions - ( A term is said to be distributed if the proposition makes an assertion about every member of the class denoted by the term; otherwise, it is undistributed)

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Venn Diagrams

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a system of diagrams to represent the information expressed (created by John Venn)

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modern square of opposition

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diagram that represents the relationship of mutually contradictory pairs of propositions

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contradictory relation

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two propositions that necessarily have opposite truths

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immediate references

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have only one premise (ex:Some trade spies are not masters at bribery. Therefore, it is false that all trade spies are masters at bribery.)

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unconditionally valid

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used to describe Arguments that are valid from the Boolean standpoint because they are valid regardless of whether or not their terms refer to existing things

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existential fallacy

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a formal fallacy that is committed whenever an argument is invalid merely because the premise is interpreted as lacking existential import (ex: 1. All A are B. Therefore, some A are B. 2. No A are B. Therefore, some A are not B.

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conversion

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it consists in switching the subject term with the predicate term. (ex: No foxes are hedgehogs & \"No hedgehogs are foxes)

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logically equivalent statements

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when two statements necessarily have the same truth value

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obversion

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requires two steps: (1) changing the quality (without changing the quantity), and (2) replacing the predicate with its term complement

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class complement

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is the group consisting of everything outside the class (ex. the complement of dogs would be: fish, trees, cats)

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term complement

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the word or group of words that denotes the class complement (ex: \"dog\" is \"non-dog)

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contraposition

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requires two steps: (1) switching the subject and predicate terms and (2) replacing the subject and predicate terms with their term complements (ex: \"All goats are animals\" is contraposed, the resulting statement is \"All non-animals are non-goats.\" )

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Law of Disjunctive Inference

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pVq ~p *q

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Definition of Conjunction

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pΛq *p *q

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Law of Detachment

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p→q p *q

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Definition of Disjunction

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p q *pVq *pVq

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Law of Contrapositive

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p→q *~q→~p

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Law of Syllogism (Chain Rule)

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p→q q→r *p→r

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Law of Modus Tollens

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p→q ~q→~p ~q p *~p *q

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DeMorgan's Law

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(pΛq)↔ ~(~pV~q) (pVq)↔ ~(~pΛ~q) ~(pΛq)↔ (~pV~q) ~(pVq)↔ (~pΛ~q)

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Law of Detachment

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p-->q p q

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DeMorgan's Law

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~(p^q) or ~(p v q) / ~p v ~q or ~p ^ ~q

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Law of Contrapositive

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p --> q / ~q --> p

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Law of Contrapositive Inference

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p --> q ~q / ~p

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Law of Disjunctive Inference \"the maid or butler did it\"

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p v b or p v b ~p or ~b / b or p

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Well-Ordering Principle

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Every nonempty set S of nonnegative integers contains a least element; that is, there is some integer a in S such that a≤b for all b's in S

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First Principle of Finite Induction

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Let S be a set of positive integers with the following properties: (a) 1 is in S (b) If k is in S , then k + 1 is in S. Then S is the set of all positive integers

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Second Principle of Finite Induction

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Let S be a set of positive integers with the following properties: (a) 1 is in S (b) If k is a positive integer such that 1, 2, . . . , k are in S , then k +1 is S. Then S is the set of all positive integers

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The Division Algorithm

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Given integers a and b, with b > 0, there exist unique integers q and r satisfying a = qb + r and 0 ≤ r < b. The integers q and r are called, respectively, the quotient and the remainder

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Euclid's Lemma

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If a l bc, with gcd(a,b) = 1, then a l c Proof: since the gcd(a,b)=1, 1=ax+by for x,y in Z so c=(ax+by)c=acx+bcy since a l ac and a l bc, a l (acx + bcy) so a l c(ax+by). Therefore a l c

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Fundamental Theorem of Arithmetic

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Every positive integer n > 1 can be expressed as a product of primes; this representation is unique, apart from the order in which the factors appear.

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Euclid's Theorem

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There are infinitely many primes

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Fermat's Theorem

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Let p be a prime and suppose that p does not divide a. Then a^p-1 ≡ 1 (mod p) Corollary: If p is a prime, then a^p ≡ a (mod p) for any integer a.

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Chinese Remainder Theorem

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Let n1, ..., nr be positive integers such that gcd(ni,nj)=1 for i≠j. The system of congruence x ≡ a1 (mod n1) . . . x ≡ ar (mod nr) has a simultaneous solution, which is unique modulo n = n1n2 . . . nr

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Commutative Laws: For all sets A & B

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(a) A U B = B U A (b) A inter B = B inter A

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Associative Laws: For all sets A,B, & C

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(a) (A U B) U C = A U (B U C) (b) (A inter B) inter C = A inter (B inter C)

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Distributive Laws: For all sets A, B, & C

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(a)A U (B inter C) = (A U B) inter (A U C) (b)A inter (B U C) = (A inter B) U (A inter C)

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Identity Laws: For all sets A

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(a)A U 0 = A (b)A inter 0 = 0

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Complement Laws

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(a)A U A^c^ = Univ (b)A inter A^c^ = 0

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Double Complement Laws: For all sets A

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(A^c^)^c^=A

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Idempotent Laws: For all sets A

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(a)A U A = A (b)A inter A = A

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Universal Bound Laws: For all sets A

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(a)A U Univ = Univ (b)A inter 0 = 0

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De Morgan's Laws: For all sets A and B

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(a)(A U B)^c^ = A^c^ inter B^c^ (b)(A inter B)^c^ = A^c^ U B^c^

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Absorption Laws: For all sets A and B

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(a)A U (A inter B) = A (b)A inter (A U B) = A

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Complements of U and 0

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(a)Univ^c^=0 (b)0^c^=Univ

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Set Different Law: For all sets A & B

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A - B = A inter B^c^

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Set

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a collection of objects

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element

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each object (member) in a set

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subset

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a set that contains some or all of the elements of a set

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union

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a set that contains all elements of both sets

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empty set

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a set that contains no elements

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intersection

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a set that contains only those elements that are in both sets

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compliment

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all the elements not in the set

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universal set

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the set containing all elements of a problem under consideration

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SET

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Collection of items ex. A={1, 2, 3, 4}

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ELEMENT

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item in a set

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UNION

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a set of all the elements of the original sets

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INTERSECTION

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set containing the elements the original sets have in common

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EMPTY SET

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contains no elements

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UNIVERSE (OR UNIVERSAL SET)

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contains all elements relating to situation

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COMPLIMENT

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set of all elements that weren't in the original set

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SUBSET

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a set contained within another set

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CROSS PRODUCT (CARTESIAN PRODUCT)

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a set where elements are ordered pairs

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The symbol for element

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∈

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The symbol for sub set

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⊂

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the symbol for intersection

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∩

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The symbol for union

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∪

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the symbol for no intersection

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∅

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the symbol for empty set

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{ }

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The symbol for complement

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Aʹ

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Elements/Members

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Objects belonging to the set; Designated in 3 ways: A Word Description, Listing Method, & Set Builder Notation.

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Empty Set/Null Set

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The set containing no elements. Symbols: ∅ or { } .

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SET

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A collection of objects.

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∈ & ∉

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Replaces the words: \"is an element of.\" & ∉ means \"not an element of\"

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Natural or Counting Numbers

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{ 1, 2, 3, 4, . . . }

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Whole Numbers

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{ 0, 1, 2, 3, 4, . . . }

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Integers

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{ . . . ,-3, -2, -1, 0, 1, 2, 3, 4, . . . }

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Rational Numbers

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{ p/q | p and q are integers and q ≠ 0 }

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Real Numbers

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Anything on the number line; { x| x is a number that can be expressed as a decimal }

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Irrational Numbers

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{ x| x is a real number that can not be expressed as a quotient of integers }

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Cardinal Number/Cardinality

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The number of elements in a set. n(A) or \"n of A,\" represents the cardinal number of set A. If they repeat in a set, they are not counted more than once.

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Finite Set

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The cardinal number of this set is a particular whole number (0 or counting number/a counting number).

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Infinite Set

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A cardinal number is not found; too large too count. ∞ (symbol).

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Set Equality

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Set A is equal to set B provided the following two conditions are met: 1. Every element of A is an element of B and 2. Every element of B is an element of A. Regardless of order.

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Non-Overlapping Venn Diagram

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\"None\"

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Overlapping Venn Diagram

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\"Some\"

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Subset Venn Diagram

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\"All\" or \"Some\"

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Converse

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Switch statement front to back

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Inverse

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Opposite of both parts of statement

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Contrapositive

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Opposite of both parts of the statement AND switch front to back

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Validity of Three Statements

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End of Statement 1 matches beginning of Statement 2. Statement 3 is made of the beginning of Statement 1 and the end of statement 2. If A, then B. If B, then C. If A, then C.

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Conditional statements (conditionals):

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If-then statements

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Biconditional statement:

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\"if and only if\" statements - p if and only if q - used if you can prove that a conditional statement and its converse are true.

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Hypothesis (p):

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The part in the \"if\"

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Conclusion (q):

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the part in the \"then\"

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If p,

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then q.

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Converse:

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If q, then p.

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Inverse:

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If not p, then not q.

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Contrapositive:

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If not q, then not p.

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Logically equivalent statements:

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- statements are logically equivalent if they are both true or both false - only true for statement and contrapositive, converse and inverse

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Compound statements:

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2 statements joined by or (∨) or and (∧)

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Conjunction:

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a compound statement joined by \"and\"

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Disjunction:

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a compound statement joined by \"or\"

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basic truth table: OR

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p q p OR q ------------ T T T T F T F T T F F F

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basic truth table: AND

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p q p AND q ------------ T T T T F F F T F F F F

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basic truth table: THEN

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p q p --> q ------------ T T T T F F F T T F F T ***Note this is only false when the conclusion is false and the hypotheses is true.

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Tautology:

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a statement that is always true (The last column of the truth table is all T's)

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Modus ponens

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B follows from A and A ⇒ B

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Generalisation

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(∀x_i)A follows from A

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Logical Axiom A1

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A ⇒ (B ⇒ A)

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Logical Axiom A2

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(A ⇒ (B ⇒ C)) ⇒ ((A ⇒ B) ⇒ (A ⇒ C))

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Logical Axiom A3

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(¬B ⇒ ¬A) ⇒ ((¬B ⇒ A) ⇒ B)

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Logical Axiom A5

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(∀x_i)(A ⇒ B) ⇒ (A ⇒ (∀x_i)B)

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Particularisation Rule A4

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If t is free for x in A(x) then (∀x)A(x)⊦A(t)

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Existential Rule E4

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Let t be a term that is free for x in a wf A(x,t), and let A(t,t) arise from A(x,t) by replacing all free occurrences of x by t. Then A(t,t) ⊦ (∃x)A(x,t)

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S1

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x = y ⇒ (x = z ⇒ y = z)

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S2

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x = y ⇒ x' = y'

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S3

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0 ≠ x'

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S4

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x' = y' ⇒ x = y

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S5

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x + 0 = x

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S6

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x + y' = (x + y)'

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S7

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x⋅0 = 0

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S8

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x⋅(y') = (x⋅y) + x

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S9 or The Principle of Mathematical Induction

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For any wf A(x) of S, A(0) ⇒ ((∀x)(A(x) ⇒ A(x')) ⇒ ((∀x)A(x))

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Proof t = t

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1. t+0 = t (S5') 2. (t+0=t) ⇒ (t+0 = t ⇒ t=t) (S1') 3. t+0 = t ⇒ t = t (1,2,MP) 4. t=t (1,3,MP)

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t < s

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(∃w)(w≠0 ∧ w + t = s)

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t ≤ s

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t < s ∨ t = s

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t > s

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s < t

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t ≥ s

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s ≤ t

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Number Theoretic Function

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A function whose arguments and values are natural numbers.

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Number Theoretic Relation

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A relation whose arguments are natural numbers.

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Expressible In K

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A number-theoretic relation R of n arguments is said to be this, if and only if there is a wf A(x_1, ..., x_n) of K with n free variables such that, for any natrual numbers k_1, ..., k_n, the following hold: 1. If R(k_1, ..., k_n) is true, then ⊦K A(bar{k}_1,..., bar{k}_n) 2. If R(k_1, ..., k_n) is false, then ⊦K ¬ A(bar{k}_1,..., bar{k}_n)

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Representable in K

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A number-theoretic function f on n arguments is said to be this, if and only if there is a wf A(x_1, ..., x_n+1) of K with the free variables x_1, ..., x_n+1 such that, for any natural numbers k_1, ..., k_n, m, the following hold: 1. If f(k_1, ..., k_n) = m then ⊦K A(bar{k}_1,..., bar{k}_n, m) 2. ⊦K (∃_1 x_n+1) A(bar{k}_1,..., bar{k}_n, x_n+1)

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Strongly Representable in K

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If f is this, then f is representable in K and 2'. ⊦K (∃_1 x_n+1) A(x_1,..., x_n, x_n+1)

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Characteristic Function

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C_R (x_1, ..., x_n) = { 0 if R(x_1, ..., x_n) is true 1 if R(x_1, ..., x_n) is false

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Initial Functions

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1. Zero Function Z(x) = 0 2. Successor Function N(x) = x + 1 for all x 3. Projection Functions U_i^n (x_1, ..., x_n) = x_i for all x_1, ..., x_n

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Primitive Recursive

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A function f is this, if and only if it can be obtained from the initial functions by any finite number of substitutions and recursions.

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Recursive

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A function f is this, if and only if it can be obtained from the initial functions by any finite number of substitutions, recursions and the μ-operator

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contrapositive

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the statement formed when you negate the hypothesis and conclusion of the converse of a conditional statement logically equivalent to conditional

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counter example

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an example that shows a conjecture is false

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proof

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the act of validating

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implication

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any proposition written in if ....then format

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premise

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a statement that is assumed to be true and from which a conclusion can be drawn

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sound

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a valid agrument with true premises

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not

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the word used to express the negation of a proposition

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antecedent

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The If clause of an implication is called the

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converse

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For any proposition p==>q, a proposition that reverses the Order of the premises to produce q===>p

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modus tollens

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a syllogism of form if p then q. Not q. Therefore not p. proves something wrong

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hypothetical syllogism

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a valid syllogism that resembles the transitive property in algebra.

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negation

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if proposition p is true then the____of P is false

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given

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a reason a proof that is not a logical tool

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if and only if

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key four word phrase indicating a biconditional

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proposition

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any statement is logic

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reduction ad absurdum

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Latin name for an indirect argument in which one temporarily assimes the negation of the desired conclusion and reasons until he reaches a contridiction of a known fact

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venn diagram

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graphic representation of an argument in logic

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and

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the key word used to indicate that a proposition is a conjunction

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or

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disjunction is indicated by the keyword

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disjunction

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a proposition that is false only when both of its premises are false

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substitution

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If angle 1 an angle 2 are supplementary and angle 2= angle 3 the angle 1 and angle 3 are supplementary.

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subset

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If all members of Group B are also members of a larger group A, then we say that Group B is a(n)_________ of Group A

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some

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the word commonly used as a principal existenial quantifier

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consequent

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the then clause of an implication

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modus ponens

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valid syllogism in the form if P the Q. P is true. Therefore Q is true

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logically equivalent

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two propositions with the same truth table values

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truth table

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a systematic arrangement of all possible truth values for a logical proposition

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jingoism

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the unwarranted, exaggerated patriotism loyalty to a person, institution, or ideal without adequate logical justification

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intersection

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a venn diagram of the logical conjunction is represented by points that lie in the __________

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conclusion

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a statement logically inferred from other statement

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transitive

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If the measure of angle k is equal j. j=37 the angle k=37 is an illustration of this property

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accentuation

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logical slight of hand. the attempt to distract the reader from the main issue by placing emphasis upon a tangential issue

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hypothesis

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an educated guess about what will be the outcome of an experiment. sometimes refered to the antecedent of an implication

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biconditional

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if both the conditional and the converse of a proposition are true then the proposition

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All

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affirmative universal quantifier

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fallacy

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an error in logic which may concern either the form of the argument or the truthfulness of its premises

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stereotyping

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logically fallacy in which one attributes distorted exaggerated characterisics of a class of items or people to a specific indivudial with out justification

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syllogism

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a set of premises whose form always produces a true conclusion

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false dilema

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artificial reduction of all possible choices to two choices either of which result in negative consequences when in truth there are better options

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truth value

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either true or false but not both

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slippery slope

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logically fallacy that infers an inevitable sequence of causes and effects such that the occurence of the first cause must produce the final effect

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inverse

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implication that negates both the antecedent and the consequent of the conditional

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amphiboly

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logical fallacy resulting from the grammatical ambiguity of sentence structure

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valid

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form guarantees a true conclusion anytime that the premises are true

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strawman

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we exaggerate

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contridiction

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indirect proof, one temporary assumes the negation of the conclusion and reasons logically until he finds a _________

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no

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most common negative universal quantifier

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hasty conclusion

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inference drawn from insufficient premises, the premises may not have accounted for all possible contributing influences leading to the inference.

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ad hominem

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logical fallacies that are an attack against the person offering an argument itself

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begged

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assumes the conclusion as a premise in the argument to produce a circular reasoning pattern

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ad misericordium

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makes a decision about an issue based upon sympathy for the issue instead of genuine qualifications for the issue

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faulty clause

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accepts the assertion that an event is the result of one factor when it is really the result of another

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Statement

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p -> q

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Converse

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q -> p

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Inverse

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~p -> ~q

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Contrapositive

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~q -> ~p

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Law of Detachment

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p -> q and p ______ .. q

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Law of Contraposition

answer

p -> q and ~q ______ .. ~p

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Law of Syllogism (transitive)

answer

p -> q q -> r p ____ .. r

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Law of Disjunctive sullogism

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p or q ~p ____ . . q

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AND

answer

p ^ q True only if both True

question

OR

answer

p V q False only if both false

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NEGATION

answer

~p opposite

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IF ... THEN

answer

p | r | if p then q T | T | T T | F | F F | T | T F | F | T

question

Implication

answer

p -- > q

question

Converse

answer

q --> p

question

Inverse

answer

~ p --> ~ q

question

Contrapositive

answer

~ q --> ~ p

question

Logically Equivalent Conditions

answer

Original (p --> q) === Contrapositive (~q --> ~p) OR Converse (q --> p) === Inverse (~p --> ~q)

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~ (p --> q) ===

answer

p ^ ~ q

question

p ^ ~ q ===

answer

~ (p --> q)

question

p -- > q (Orig) ===

answer

~ q --> ~ p (Contrapositive)

question

q --> p (Converse) ===

answer

~ p --> ~ q (Inverse)

question

~ q --> ~ p (Contrapositive) ===

answer

p -- > q (Orig)

question

~ p --> ~ q (Inverse) ===

answer

q --> p (Converse)

question

Conditional: If today is Easter, then tomorrow is Monday (write the Contrapositive)

answer

Contrapositive: If tomorrow is not Monday, then today is not Easter

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Negation of p --> q

answer

p ^ ~ q

question

Conditional: If today is Easter, then tomorrow is Monday (write the Converse)

answer

Converse: If tomorrow is Monday, then today is Easter.

question

Conditional: If today is Easter, then tomorrow is Monday (write the Inverse)

answer

Inverse: If today is not Easter, the tomorrow is not Monday.

question

Conditional: If Tom is Ann's father, then Jim is her uncle and Sue is her aunt. (write Contrapositive)

answer

Contrapositive: If either Jim is not Ann's uncle or Sue is not her aunt, then Tom is not her father

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Conditional: If X, then Y AND Z

answer

Contrapositive: If EITHER not Y OR not Z, then not X

question

Conditional: If X, then Y OR Z

answer

Contrapositive: If not Y AND Z, then not X

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Modus Ponens

answer

p --> q p ... q [VALID] modus ponendo ponens: \"the way that affirms by affirming\"

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Modus Tollens

answer

p --> q ~ q ... ~ p [VALID] modus tollendo tollens: \"the way that denies by denying\"

question

Elimination

answer

p v q ~ p ... q [VALID]

question

Transitivity

answer

p --> q q --> r ... p --> r [VALID]

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Generalization

answer

p ... p v q

question

Specialization

answer

p ^ q ... p

question

Proof by Division into cases

answer

p v q p --> r q --> r ... r [VALID]

question

Conjunction

answer

p q ... p ^ q

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Contradiction

answer

~ p --> C ... p

question

Converse Error

answer

p --> q q ... p [INVALID]

question

Inverse Error

answer

p --> q ~ p ... ~ q [INVALID]

question

p OR q ===

answer

~ p --> q

question

DeMorgans Laws

answer

~(p ^ q) === ~p v ~q ~(p v q) === ~p ^ ~q

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p --> q ===

answer

~ p v q

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p --> q ===

answer

~q --> ~p

question

p q ===

answer

(p --> q) ^ (q --> p) (p ^ q) v (~p ^ ~q)

question

Vacuously True

answer

Hypothesis is False

question

Implication is False

answer

ONLY when hyp (p) is T and conc (q) is F (p(T) --> q(F))

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Law of Detachment

answer

P implies Q, P is true ... Q is true

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Chain Rule/Law of Syllogism

answer

P implies Q, Q implies R ... P implies R/If..and..then

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Biconditional

answer

PQ, P is true ... Q is true / If and only if

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Contraposition

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P implies Q, Q is false ... P is false

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Disjunctive Inference

answer

P or Q, P is not true ... Q is true

question

Denying the Premise/Hypothesis

answer

P implies Q, P is false ... Q is false

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Asserting the Conclusion

answer

P implies Q, Q is true ... P is true

question

True conjunction

answer

A conjunction is true if and only if both conjunct are true

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True Biconditional

answer

(p→q) ∧ (q→p).

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Law of disjunctive inference

answer

If a disjunction is true and one of its disjuncts is false, then the other disjunct must be true. [(p ∨ q) ∧ ~p] → q

question

Law of detachment

answer

If the antecedent of a true conditional is true then the consequent of the same conditional must be true. [(p → q) ∧ p ]→ q

question

Law of double negation

answer

p → ~(~p)

question

Law of the contraposative

answer

A contraposative of a true conjunction has the same meaning as the original conjunction. (p → q) ↔ (~q → ~p)

question

Converse

answer

q → p

question

Contraposative

answer

~q → ~p Inverse + Converse

question

Inverse

answer

~p → ~q

question

Replacement Set

answer

The domain

question

Law of Modus Tollens

answer

If the consequent of a true conditional statement is false, then the antecedent must be false. [(p → q) ∧ ~p] → ~q

question

Law of Syllogism

answer

If (p → q) and (q → r) are true, then (p → r) is true

question

Demorgan's Law

answer

~(p ∧ q) ↔ (~p ∨ ~q) ~(p ∨ q) ↔ (~p ∧ ~q)

question

QED!

answer

which was to be demonstrated (latin)

question

Disjunctive addition

answer

p → (p ∨ q)

question

Disjunctive equivalent of a conditional

answer

Every conditional has a disjunctive equivalent (p → q) ↔ (~p ∨ q)

question

Negation of Conditionals

answer

~(p → q) ↔ ~(~p ∨ q) can use demorgans law to find: ~(p → q) ↔ ~(~p) ∧ ~q then law of double negation: ~(p → q) ↔ p ∧ ~q

question

Conjunctive Addition

answer

If two conjuncts are true, their conjunction will be true. p ∧ q → (p ∧ q)

question

Given the statements p and q, an implication is a statement that is false when p is true and q is false, and true otherwise.

answer

→

question

A biconditional statement is true whenever the truth value is the same for both p and q and false otherwise.

answer

↔

question

'NOT:'Negation - a method of assigning the opposite truth value to the statement.

answer

¬

question

AND:' Given p and q, a conjunction is the proposition that is true when both p and q are true and is false otherwise.

answer

∧

question

'OR:' Given p and q, a disjunction is the proposition that is false when both p and q are false, but is true otherwise.

answer

∨

question

'XOR:' An exclusive or is a proposition which is true when exactly one of p and q is true and is false otherwise.

answer

⊕

question

The universal quantification of P(x) is the proposition \"P(x) is true for all values x in the universe of discourse.\"

answer

∀

question

The existential quantification of P(x) is the proposition \"There exists an element x in the universe of discourse such that P(x) is true.\"

answer

∃

question

conditional statement

answer

a statement in the form of \"if p, then q\"

question

converse statement

answer

the converse of a conditional statement is formed by negating the hypothesis and the conclusion to become \"if q, then p\"

question

inverse statement

answer

the inverse of a conditional statement is formed by negating the hypothesis and the conclusion to become \"if not p, the not q\"

question

contrapositive statement

answer

the contrapositive of a conditional statement is formed by reversing AND negating the hypothesis and the conclusion to become \"if not q, then not p\"

question

counterexample

answer

shows the hypothesis (if part) of a conditional can be true without the conclusion (then part) also being true

question

biconditional

answer

a condition whose converse is also true. (denoted with a double ended arrow)

question

proportion

answer

an equation showing that two ratios are equivalent

question

ratio

answer

a comparison of two or more numbers using division

question

similar

answer

having exactly the same shape, but not necessarily the same size. similar figures have proportional corresponding sides and congruent corresponding angles

question

(triangle) side-angle inequality

answer

the longest side is opposite the largest angle and the smallest side is opposite the smallest angle

question

triangle inequality

answer

any side of a triangle must be greater than the difference and smaller than the sum of the other two sides

question

Ad Hominem

answer

The fallacy of personal attack; debater introduces irrelevant personal facts about his opponent

question

Ad Miscordium

answer

An appeal to pity; someone tries to win support for their argument or idea by exploiting his or her opponents feelings of pity or guilt

question

All

answer

universal quantification; the notion that something is true for everything (or every relevant thing)

question

Amphiboly

answer

Fallacy of ambiguation: involves the use of sentences which can be interpreted in multiple ways with equal justification

question

And

answer

a two place logical connective that has the value true if both operants are true

question

Antecedent

answer

The first half of a hypothetical situation. Follows \"if\"

question

Begged (the question)

answer

When one assumes his conclusion as a premise in his argument to produce a circular reasoning pattern

question

Biconditional

answer

logical operator connective two statements to assert \"P if an only Q\"

question

Conclusion

answer

\"logical consequence\";the relationship between the premises and the conclusion of a valid argument. It is the relation that holds between a set of propositions and another proposition when the former imply the latter. EX. \"Kermit is green\" is the logical consequence of \"all frogs are green\"

question

Conjunction

answer

(same as and) a two-place logical connective that has the value true if both of its operants are true

question

Consequent

answer

The second half of a hypothetical proposition. The part that follows 'then'

question

Contradiction

answer

logical incompatibility between two or more propositions. When two propositions yield conclusions that are logical inversions of each other

question

Contrapositive

answer

the statement formed when you negate the hypothesis and conclusion of the converse of a conditional statement; if not \"p then not q\"

question

Converse

answer

the statement formed by switching the hypothesis and conclusion of a conditional statement; \" No Romans are Philosophers\" = \"No Philosophers are Romans\"

question

counter example

answer

An exception to a proposed general rule that proves it false.

question

disjunction

answer

logical operator that results in true when one or more of its operands are true

question

Fallacy

answer

misconception resulting from false reasoning in logical argumentation

question

False dilemma

answer

a situation in which only two alternatives are considered, when in fact there are other options

question

Faulty Cause (-and effect)

answer

Correlation does not prove causation

question

given

answer

an assumption that is taken for granted

question

hasty conclusion

answer

inference drawn from insufficient premises, the premises may not have accounted for all possible contributing influences leading to the inference.

question

hypothesis

answer

a proposal intended to explain certain facts or observations

question

hypothetical syllogism

answer

\" rule of inferenece\";If one implies the other, and that other implies a third, the first implies the third.

question

If and only If

answer

used with the biconditional

question

Implication

answer

the \"If\" clause; relationship between two propositions where the truth of one requires the truth of the other.

question

inverse

answer

the statement formed when you negate the hypothesis and conclusion of a conditional statement

question

jingoism

answer

A fallacy of irrelevancy; unwarranted, often exaggerated, patriotism or loyalty to a person , institution, or ideal without adequate logical justification

question

Logically Equivalent

answer

possessing the same logical content. \"If lisa is in France, she is in Europe\" = \"If Lisa is not in Europe, she is not in France.\"

question

Modus Ponens

answer

affirming the antecedent; if the weather forecast calls for rain ill need the umbrella. The forcast calls for rain. Therefore, ill need my umbrella

question

modus tollens

answer

denying the consequent; if i have strep throat, then the culture will be positive. The culture was not positive. Therefore i dont have strep throat

question

negation

answer

(logic) a proposition that is true if and only if another proposition is false

question

no

answer

a negative

question

not

answer

Expresses Logical negation; Takes the truth to fallacy. Usually precedes a statement.

question

or

answer

Expresses Logical Disjunction; Results in true when one or more of its operands are true.

question

premise

answer

a statement that is assumed to be true and from which a conclusion can be drawn

question

Proof

answer

logical argument that shows a statement is true

question

Proposition

answer

any statement in logic

question

reductio ad absurdum

answer

disproof by showing that the consequences of the preposition are absurd or contradicting

question

Slippery Slope

answer

Fallacy; states that a relatively small first step inevitably leads to a chain of related events culminating in some significant impac

question

some

answer

term associated with the existential qualifier

question

Existential Qualifier

answer

property or relation holds true to at least one member of the domain

question

stereotyping

answer

attributing distorted or exaggerated characteristics of a class of items or people to a specific individual without proper justification.

question

strawman

answer

logical fallacy based on misrepresentation of an opponents position

question

subset

answer

a set whose members are members of another set

question

syllogism

answer

deductive reasoning in which a conclusion is derived from two premises

question

truth table

answer

a systematic arrangement of all possible truth values for a logical proposition

question

truth value

answer

whether a statement is true or false

question

valid

answer

logically convincing; sound; legally acceptable; effective; Ex. valid reasoning/passport

question

venn diagram

answer

A diagram that shows relationships among sets of objects.

question

Symbol for intersection

answer

∩

question

Symbol for union

answer

∪

question

Symbol for proper subset (strictly smaller than)

answer

⊂

question

Symbol for subset (equal to or smaller than)

answer

⊆

question

Complement of set A (objects not in set A)

answer

A'

question

Symbol for conjunction

answer

∧

question

Symbol for disjunction (note: you need to know exclusive disjunction too)

answer

∨

question

Symbol for negation (\"not\")

answer

⌉

question

Symbol for element of a set

answer

∈

question

finite elements in the set (not infinite)

answer

finite set

question

infinite (endless) elements in the set (e.g. the set of real numbers is infinite)

answer

infinite set

question

the set encompassing all elements

answer

universal set

question

all elements in the universe that is not in A

answer

complement of set A

question

Symbol for the null (empty) set

answer

∅

question

A set and its complement do not intersect (it is impossible for an element to be in a set and its complement)

answer

A∩A'=∅

question

A set and its complement is equal to the universal set

answer

A∪A'=U

question

the size (number of elements) of the set A

answer

n(A)

question

the size of A equals the size of the universe minus everything not in A

answer

n(A) = n(U)-n(A')

question

Double Negation

answer

∼(∼p) ∴(p)

question

Definition of Conjunction

answer

p, q ∴(p ∧ q)

question

Conjunctive Simplification

answer

p ∧ q ∴(p)

question

Disjunctive Addition

answer

p ∴(p ∨ q)

question

Disjunctive Inference

answer

p ∨ q, ∼q ∴(p)

question

De Morgan's

answer

∼(p ∨ q) ∴(∼p ∧ ∼q)

question

Chain Rule

answer

p → q, q → r ∴(p → r)

question

Detachment

answer

p → q, p ∴(q)

question

Modus Tollens

answer

p → q, ∼q ∴(∼p)

question

Material Implication

answer

p → q ∴(∼p ∨ q)

question

Alternate Exterior Angles

answer

angles that lie outside the two lines on opposite sides of a transveral

question

Alternate Interior Angles

answer

angles that lie between two lines on opposite sides on the transversal

question

Biconditional

answer

conditional statement and converse

question

Conclusion

answer

comes after \"then\"

question

Conditional

answer

statement in \"if\" \"then\" format

question

Conjecture

answer

conclusion reached by using inductive reasoning

question

Consecutive Int.

answer

angles that lie between two lines on the same side of the transveral

question

Contrapositive

answer

switches and negates the hypothesis and the conclusion

question

Converse

answer

switches the hypothesis and conclusion

question

Corrseponding Angles

answer

angles that lie in matching positions

question

Counterexample

answer

shows conditonal if false (counter the conclusion)

question

Deductive Reasoning

answer

process of reasoning logically from given statements to a conclusion

question

Hypothesis

answer

comes after \"if\"

question

Inducitve Reasoning

answer

reasoning based on patterns

question

Inverse

answer

negates the hypothesis

question

Negation

answer

not, denial of the statement

question

Transversal

answer

a line that intersects two or more coplanar lines at different points

question

Truth Value

answer

true or false

CLEP College Mathematics – Mathematical Logic and Set Theory

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