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Math 3330 Section 1. Relations
A binary relation (also called simply a relation) on a nonempty set A is a
nonempty subset R of A × A.
If ( ,a b ) ∈ R , we write aRb and say “a is related to b”.
If ( ,a b ) ∉ R , then we write a Rb and say “a is not related to b”.
Any mapping on A is an example of a relation. But, a relation DOES NOT HAVE TO BE a mapping.
Example: A={1,3,5,7} and B={2,4,5,6} define R
Properties of Relations Reflexive
Irreflexive
Symmetric
Asymmetric
Anti-symmetric
Transitive
Examples of Relations Consider the set of all human beings. Let x and y be human beings.
Suppose xRy if and only if x and y live less than 100 miles apart.
Suppose xRy if and only if x and y were born the same year.
Consider the power set P(A) for a nonempty set A, and x and y are elements of P(A).
Suppose xRy if and only if x is a subset of y.
Special Relations Equivalence Relations (IMPORTANT!) A relation R is called an equivalence relation if R is reflexive, symmetric and transitive.
Examples of equivalence relations The = sign – most important
aRb if and only if a-b is a multiple of 3.
Consider the set of ordered pairs of integers (a,b) with b nonzero.
Suppose ( ,a b R c d ) ( , ) if and only if ad = bc
Show that R is an equivalence relation.
Equivalence Classes
Let R be an equivalence relation on the nonempty set A. For each define the equivalence class of a to be
a ∈ A
[ ]a = { x ∈ A : xRa}
Equivalence Classes for aRb if and only if a-b is a multiple of 3.
