## Relations & Functions

# Relation

- Let A, B be any two non-empty sets, then every subset of A × B defines a relation from A to B and every relation from A to B is a subset of A × B.
- The Image of an element x under a relation R is given by y, where (x,y) ∈ R.
- Domain of a relation R ⊆ A × B: Domain of R = {a ∈ A / (a, b) ∈ R} i.e. The set of all first elements of all order pairs in the relation R.
- Range of a relation R ⊆ A × B: Range of R = {b ∈ B / (a, b) ∈ R} i.e. The set of all second elements of order pairs in the relation R.
- If R and S are two relations from A to B, then R ∪ S, R ∩ S and R − S, are also relations from A to B.

### View the Topic in this video From 12:52 To 48:28

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

- The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A × B. If n(A) = m and n(B) = n, then n(A × B) = mn and the total number of relations is 2
^{mn}. - The number of relations that can be defined on the set A is 2^{n(A)^{2}}.