MATH3200 - Exam 2

set containment (A ⊆ B)

set containment (A ⊆ B)

A set A is a subset of a set B if every element of A is also an element of B, denoted as A⊆B.

set equality (A = B)

Two sets A and B are equal if they contain exactly the same elements, i.e., A=B if A⊆B and B⊆A.

the empty set ∅

The empty set is the set that contains no elements, denoted by ∅.

A∪B

The set of elements that are in either A or B or in both.

A∩B

The set of elements that are in both A and B.

A\B

The set of elements that are in A but not in B.

Commutative Property

For union and intersection, A∪B=B∪A and A∩B=B∩A

Associative Property

For union and intersection, (A∪B)∪C=A∪(B∪C) and (A∩B)∩C=A∩(B∩C).

Distributive Property

For sets, A∩(B∪C)=(A∩B)∪(A∩C) and A∪(B∩C)=(A∪B)∩(A∪C).

de Morgan’s laws

(A∪B)^c=A^c∩B^c and (A∩B)^c=A^c∪B^c, where c denotes the complement.

set product A × B

The Cartesian product of two sets A and B is the set of all ordered pairs (a,b) where a∈A and b∈B, denoted by A × B.

the power set P(S)

The power set of a set S is the set of all subsets of S, denoted by P(S).

Union ∪_i∈I A_i, where I is an index set

The set of elements that belong to at least one of the sets Ai​, where i ranges over the index set I.

Intersection ∩_i∈I A_i, where I is an index set

The set of elements that belong to all sets Ai​, where i ranges over the index set I.

relation on sets S and T

A relation from set S to set T is a subset of the Cartesian product S×T, i.e., a set of ordered pairs (s,t) where s∈S and t∈T.

relation on a set S

A relation on a set S is a subset of S×S, i.e., a set of ordered pairs (s₁​,s₂​) where both s₁​ and s_2​ are elements of S.

domain of a relation

The set of all first elements (or inputs) of the ordered pairs in a relation.

range of a relation

The set of all second elements (or outputs) of the ordered pairs in a relation.

reflexive

A relation R on a set S is reflexive if for every element a∈S, (a,a)∈R.

symmetric

A relation R on a set S is symmetric if for every pair (a,b)∈R, (b,a)∈R.

transitive

A relation R on a set S is transitive if whenever (a,b)∈R and (b,c)∈R, then (a,c)∈R.

equivalence relation

A relation R on a set S is an equivalence relation if it is reflexive, symmetric, and transitive.

equivalence class [x]

The equivalence class of an element x in a set S under an equivalence relation R is the set of all elements in S that are related to x, denoted by [x]={y∈S:(x,y)∈R}.

natural numbers

Numbers used for counting: 1, 2, 3, 4

Integers

All whole numbers and their negative counterparts: -2, -1, 0, 1, 2

rational numbers

Any number that can be expressed as a fraction: 1/2, .5, -3/4

real numbers

All numbers that can be found on the number line, rational or irrational: 7, -1.2, sqrt(55), e