Studied by 0 people
Looks like no one added any tags here yet for you.
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=Ac∩Bc and (A∩B)c=Ac∪Bc, 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 Ai, 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 Ai, 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 s2 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
