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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CBSE Class 11 Maths Sets Relations and Functions All Topic Notes CBSE Class 11 Maths All Chapters Notes