# Binary Relation

Let P and Q be two non- empty sets. A binary relation R is defined to be a subset of P x Q from a set P to Q. If (a, b) âˆˆ R and R âŠ† P x Q then a is related to b by R i.e., aRb. If sets P and Q are equal, then we say R âŠ† P x P is a relation on P e.g.

**Example1:** If a set has n elements, how many relations are there from A to A.

**Solution:** If a set A has n elements, A x A has n^{2} elements. So, there are 2^{n2} relations from A to A.

**Example2:** If A has m elements and B has n elements. How many relations are there from A to B and vice versa?

**Solution:** There are m x n elements; hence there are 2^{m x n} relations from A to A.

**Example3:** If a set A = {1, 2}. Determine all relations from A to A.

**Solution:** There are 2^{2}= 4 elements i.e., {(1, 2), (2, 1), (1, 1), (2, 2)} in A x A. So, there are 2^{4}= 16 relations from A to A. i.e.

## Domain and Range of Relation

**Domain of Relation:** The Domain of relation R is the set of elements in P which are related to some elements in Q, or it is the set of all first entries of the ordered pairs in R. It is denoted by DOM (R).

**Range of Relation:** The range of relation R is the set of elements in Q which are related to some element in P, or it is the set of all second entries of the ordered pairs in R. It is denoted by RAN (R).

**Example:**

**Solution:**

DOM (R) = {1, 2} RAN (R) = {a, b, c, d}

## Complement of a Relation

Consider a relation R from a set A to set B. The complement of relation R denoted by R is a relation from A to B such that

` R = {(a, b): {a, b) âˆ‰ R}. `

**Example:**

**Solution:**

X x Y = {(1, 8), (2, 8), (3, 8), (1, 9), (2, 9), (3, 9)} Now we find the complement relation R from X x Y R = {(3, 8), (2, 9)}