Chapter 1 Relations and Functions

Notes

Relations and Functions – Complete Notes

1. Relations

Let A and B be two non-empty sets. A relation R from A to B is a subset of the Cartesian product A × B.

R ⊆ A × B

If a relation is defined on a set A, then:

R ⊆ A × A

A relation is represented by a collection of ordered pairs (a, b).

1.1 Example of a Relation

Let A = {1, 2, 3} and define:

R = {(a, b) : a > b, a, b ∈ A}

Then:

R = {(2, 1), (3, 1), (3, 2)}

If the relation is defined by a < b, then:

R = {(1, 2), (1, 3), (2, 3)}

Every ordered pair of a relation must belong to the given Cartesian product.

2. Domain and Range of a Relation

• Domain: The set of all first components of the ordered pairs belonging to R.
• Range: The set of all second components of the ordered pairs belonging to R.

Example:

R = {(1, 3), (2, 6), (3, 9)}

Domain = {1, 2, 3}

Range = {3, 6, 9}

3. Number of Relations

Let n(A) = m and n(B) = n.

The number of elements in A × B is:

n(A × B) = mn

Since every subset of A × B is a relation, the total number of relations from A to B is:

2^(mn)

4. Types of Relations

4.1 Reflexive Relation

A relation R defined on a set A is called reflexive if every element is related to itself.

(a, a) ∈ R for every a ∈ A

Example:

Let A = {a, b, c}.

R = {(a, a), (b, b), (c, c)}

R is reflexive because every element of A is related to itself.

4.2 Symmetric Relation

A relation R defined on A is called symmetric if:

(a, b) ∈ R ⇒ (b, a) ∈ R

Example:

R = {(a, b), (b, a)}

This relation is symmetric.

However, R = {(a, b), (b, c)} is not symmetric if the corresponding reverse pairs are absent.

4.3 Transitive Relation

A relation R is called transitive if:

(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R

Example:

R = {(a, b), (b, c), (a, c)}

This relation is transitive.

5. Equivalence Relation

A relation R defined on a set A is called an equivalence relation if it is:

1. Reflexive
2. Symmetric
3. Transitive

All three conditions must be satisfied.