# Relations Formula

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## Ordered Pair

An ordered pair consists of two objects or elements grouped in a particular order,

### Equality of ordered Pairs

Two ordered pair (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2

### Cartesian Product of Sets

For two non-empty sets A and B, the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B is called Cartesian Product A × B, i.e.

Suppose A = {a, b} and B = {p, q}

A × B = {a, b} × {p, q}

= {(a, p) (a, q) (b, p) (b, q)}

### Properties of Cartesian Product

1. n(A × B) = n(A) × n(B)
2. A × (B ∪ C) = (A × B) ∪ (A × C)
3. A × (B ∩ C) = (A × B) ∩ (A × C)
4. A × (B – C) = (A × B) – (A × C)
5. (A × B) ∩ (C × D) = (A ∩ B) × (C ∩ D)
6. A × (B’ ∪ C’)’ = (A × B) ∩ (A × C)
7. A × (B’ ∩ C’)’ = (A × B) ∪ (A × C)

## Relation

If A and B are two non-empty sets, then a relation R from A to B is a subset of A × B.

If R ⊆ A × B and (a, b) ∈ R, then we say that a is related to b by the relation R, Written as aRb.

### Representation of a relation

1. Rooster form
2. Builder form
3. Arrow diagram

### Domain, Codomain, and Range of a Relation

Domain: The set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R.

Codomain: The set of all second components or coordinates of the ordered pairs belonging to R is called the range of R. Also, the set of B is called the codomain of relation R.

Thus, domain of R = {a: (a, b) ∈ R} and range of R = {b: (a, b) ∈ R}

### Types of Relations

1. Reflexive Relation: A relation R on a set A is said to be a reflexive relation if every element of A is related to itself. i.e. (a, a) ∈ R
2. Symmetric Relation: A relation R on a set A is said to be a symmetric relation iff (a, b) ∈ R and (b, a) ∈ R.
3. Transitive Relation: A relation R on a set A is said to be a transitive relation, Iff (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
4. Equivalence Relation: A relation R on a set A is said to be an equivalence relation if it is simultaneously reflective, symmetric and transitive on A.

## Inverse Relation

IF A and B are two non-empty sets and R be a relation from A to B, then the inverse of R, denoted by R-1, is a relation from B to A and is defined by R-1 = {(b, a) : (a, b) ∈ R}.

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