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Let A = {*a*, *b*}. List all relations on A and find their number.

*A*can be written as a set of ordered pairs.

The only ordered pairs that can be included are (

*a*,

*a*), (

*a, b*), (

*b, a*) and (

*b, b*).

There are four ordered pairs in the set, and each subset is a unique combination of them.

Each unique combination makes different relations in A.

{ } [the empty set]

{(

*a, a*)}

{(

*a, b*)}

{(

*a, a*), (

*a, b*)}

{(

*b, a*)}

{(

*a, a*), (

*b, a*)}

{(

*a, b*), (

*b, a*)}

{(

*a, a*), (

*a, b*), (

*b, a*)}

{(

*b, b*)}

{(

*a, a*), (

*b, b*)}

{(

*a, b*), (

*b, b*)}

{(

*a, a*), (

*a, b*), (

*b, b*)}

{(

*b, a*), (

*b, b*)}

{(

*a, a*), (

*b, a*), (

*b, b*)}

{(

*a, b*), (

*b, a*), (

*b, b*)}

{(

*a ,a*), (

*a, b*), (

*b, a*), (

*b, b*)}

Number of elements in the Cartesian product of

*A*and

*A*= $2\times 2=4\phantom{\rule{0ex}{0ex}}$

∴ Number of relations = ${2}^{4}=16$

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