IB Math HL Option: Sets, Relations and Groups (Paper 3)
Unlock all answers in this set
Unlock answersquestion
Finite set
answer
A collection of finite objects, called the element or members of the set
question
Infinite set
answer
A collection of infinite objects, called the element or members of the set
question
Power set
answer
The set of subsets of a set A called the power set. The number of subsets of a set with m elements is 2^m
question
Relation
answer
I any set of ordered pairs
question
Reflexive relation
answer
xRx
question
Symmetric relation
answer
If xRy then yRx
question
Transitive relation
answer
If xRy and yRz implies xRz
question
Equivalence
answer
Reflexive, symmetric and transitive
question
Congruence modulo n
answer
For integers x and y and positive integer n, x and y are congruent modulo n is (x-y) is an integer multiple of n.
question
Injections
answer
one to one f(a1)=f(a2) a1 = a2
question
Surjections
answer
f(a)=b A function whose range is the whole of its codomain.
question
Bijections
answer
both injective and surjective
question
Domain
answer
A->B A
question
Codomain
answer
A->B B
question
Commutative Law
answer
g1*g2 = g2*g1
question
Abelian group
answer
A group whose binary operation is commutative over the elements of the group. Any cyclic group is abelian.
question
Latin square
answer
each element appears exactly once in each row and once in each column. The latin square property is necessary but not sufficient condition for a cayley table to present a group.
question
Klein 4-group
answer
The group structure of a four element group, each element of which is self-inverse.
question
Order
answer
The order of a group is the number of elements in that group. The order of an element a is the least positive integer n such that a^n = e, the identity element.
question
homomorphism
answer
a function mapping elements of one group to another group which preserves group operations.
question
Equivalence class
answer
The set of all elements equivalent to x.
question
one step subgroup test
answer
a*b^-1 is in the set of H, H does not equal 0.
question
Dihedral group of order n
answer
The group of symmetries of a regular n-gon, usually denoted Dn. D3 is the six-element group for the symmetries of an equilateral triangle: rotations by 0, 120 and -120 degrees, and reflections through each of the symmetry lines connecting a vertex to the midpoint of the opposite side.
