CBSE Class 11 Maths Notes: Relations and Functions – Relations

Let A and B be two non-empty sets then every subset of A x B defines a relation from A to B and every relation from A to B is subset of A x B.

Let R ⊆ A x B and ≤ a, b) ∈ R. then we say that a is related to b by the relation R and write it as a R b. If ≤ a, b) ∉ R, we write it as a      b.

Example

Let A {1, 2, 3, 4, 5}, B = {1, 3}
We set a relation from A to B as: a R b iff a £ b; a ∈ A, b ∈ B. Then
R = {≤ 1, 1), ≤ 1, 3), ≤ 2, 3), ≤ 3, 3)}⊂AxB 

Domain and Range of a Relation:

Let R be a relation from A to B, that is, let R ⊆ A x B. Then

Domain R = {a: a ∈ A, ≤ a, b) ∈ R for some b ∈ B}

i.e. domain of R is the set of all the first elements of the ordered pairs which belong to R.

Also Range R = {b: b ∈ B, ≤ a, b) ∈ R for some a ∈ A},

i.e. range R is the set of all second elements of the ordered pairs which belong to R.

Thus Dom. R ⊆ A, Range R ⊆ B.

Total Number of Distinct Relations from A to B:

Suppose the set A has m elements and the set B has n elements. Then the product set A x B i.e. P ≤ A x B) will have 2^mn elements. A x B has 2^mn different subsets which are different relations from A to B.

Inverse Relation:

Let R ⊆ A x B be a relation from A to B. Then inverse relation R^–1 ⊆ B x A is defined by

R^–1 = {≤ b, a): ≤ a, b) ∈ R, a ∈ A, b ∈ B}. It is clear that

• a R b ↔ b R^–1 a
• dom R^–1 = range R and range R^–1 = dom R
• ≤ R^–1)^–1 = R

Example: Let A = {1, 2, 3, 4}, B = {a, b, c} and R = {≤ 1, a), ≤ 1, c), ≤ 2, a)}. Then
i) dom R = {1, 2}, range R = {a, c}
ii) R^–1 = {≤ a, 1), ≤ c, 1), ≤ a, 2)}

Compositions of Relations:

Let R ⊆ A x B, S ⊆ B x C be two relations. Then compositions of the relations R and S denoted bySoR⊆AxCand is defined by ≤ a, c) ∈ ≤ S o R) iff $ b ∈ B such that ≤ a, b) ∈ R, ≤ b, c) ∈ S.

Example:

Let A = {1, 2, 3}, B = {a, b, c, d}, C = {a, b, g}
R ≤ ⊆ A x B) = {≤ 1, a), ≤ 1, c), ≤ 2, d)}
S ≤ ⊆ B x C) = {≤ a, a), ≤ a, g), ≤ c, b)}
Then S o R≤ ⊆ A x C) = {≤ 1, a), ≤ 1, g), ≤ 1, b)}
One should be careful in computing the relation R o S. Actually S o R starts with R and R o S starts with S. In general S o R ≠ R o S
Also ≤ S o R)^–1 = R^–1 o S^–1, known as reversal rule

Relations in a Set:

Let R be a relation from A to B. If B = A, then R is said to be a relation in A. Thus relation in a set A is a subset of A x A.

Identity Relation:

R is an identity relation if ≤ a, b) ∈ R iff a = b, a ∈ A, b ∈ A. In other words, every element of A is related to only itself.

Universal Relation in a Set:

Let A be any set and R be the set A x A, then R is called the Universal Relation in A.

Void Relation in a Set:

ϕ is called Void Relation in a set.

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