If A and B are two non-empty sets then any subset of ordered pairs of a cartesian product A × B is called a **relation** from set A to set B. A relation is denoted by xRy or
simply R, if (x, y) ∈ R. A relation from set A
to A is called a relation on A.

**Ways of Representing
a Relation**

Relations can be expressed in various ways. Here are some
examples:

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**Examples:**

1. __By specifying or displaying a set of ordered pairs__:

(a) R = {(1, 1), (2,
4), (3, 9), (4, 16), (5, 25)}

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

2. __By standard description, using a rule or a formula__:

(a) Let A = {1, 2, 3}
and B = {1, 4, 9}. R = {(x, y): y = x^{2}}. Then R = {(1, 1), (2, 4),
(3, 9)}.

(b) Let A = {1, 2, 3}
and B = {1, 2, 3}. R = {(x, y): x = y}. Then R = {(1, 1), (2, 2), (3, 3)}

3. __By table__, such as:

(a) R = {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}

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

4. __By arrow diagram__:

R = {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}

5. __By graphs__:

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

**Domain and Range of a
Relation**

The **domain** of a
relation is the set of all the first elements of the ordered pairs of R, and its **range** is the set of all second
elements.

**Examples:**

1. Let R_{1} = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3),
(3, 3)}

Domain of R_{1}
= {1, 2, 3}

Range of R_{1}
= {1, 2, 3}

2. Let R_{2} = {(1, 2), (3, 4), (5, 6)}

Domain of R_{2}
= {1, 3, 5}

Range of R_{2}
= {2, 4, 6}

**Types of Relation**

Let us consider A = {1, 2, 3}

Then the cartesian product A × A = {(1, 1), (1, 2), (1, 3), (2,
1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

**(i) ****Reflexive Relation**

A relation R is called reflexive if every element of the
relation is related to itself. It is written as xRx. For example: R_{1}
= {(1, 1), (2, 2), (3, 3)}

**(ii) ****Symmetric Relation**

In a relation, if the first and second components of the ordered
pairs are interchanged, the relation still holds. It is called the symmetric
relation. For example: R_{2} = {(1, 2), (2, 1)}. It is written as if
xRy then yRx. If x = y then the relation is Anti-symmetric. e.g. R_{3}
= {(1, 2), (1, 3), (2, 1), (3, 1)}

**(iii) ****Transitive Relation**

A relation R is called transitive if aRb and bRc gives aRc.
Thus, if a is related to b and b is related to c then a is related to c. For
example: R_{3} = {(1, 2), (2, 3), (1, 3)}.

**(iv) ****Equivalence Relation**

A relation is called equivalence if and only if it is reflexive,
symmetric and transitive. For example: R_{4} = {(1, 1), (2, 2), (3, 3),
(1, 2), (1, 3), (2, 1), (2, 3), (3, 2), (3, 1)}

**Inverse Relation**

A relation obtained by interchanging the first and second
elements in the ordered pairs of a relation is known as the **inverse** of the given relation. If R
denotes a given relation its inverse is denoted by R^{-1}.

In symbols, if R = {(a, b): a ∈ A, b ∈ B}, then R^{-1} = {(b, a): b ∈ B, a ∈ A}.

**Worked Out Examples**

** **

**Example 1:** Let A = {1, 2, 3, 4} and B = {1, 3, 5}. Find the relation R from A
to B determined by the condition ‘x < y’.

** **

**Solution:** Here,

A = {1, 2, 3, 4} and B = {1, 3, 5}

Since x ∈ A, y ∈ B and x < y, R = {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5),
(4, 5)}

**Example 2:** Find the domain, range and inverse of the relation R = {(1, 2), (2,
4), (3, 6), (4, 8)}.

** **

**Solution:** Here,

R = {(1, 2), (2, 4), (3, 6), (4, 8)}

Domain of R = {1, 2, 3, 4}

Range of R = {2, 4, 6, 8}

R^{-1} = {(2, 1), (4, 2), (6, 3), (8, 4)}

** **

**Example 3:** In the universe R × R (R is the set of all real numbers), graph (or
sketch) the relations expressed below:

(a) R_{1} = {(x, y) : x = 0}

(b)R_{2} = {(x, y) :
y = 2x}

(c) R_{3} = {(x, y) : x < 0}

(d)R_{4} = {(x, y) :
x > 0, y > 0}

** **

**Solution:**

The graph of R_{1}, R_{2}, R_{3} and R_{4}
are shown in figures a, b, c, and d respectively.

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