Question

If A=(1,2,3) then find the number of reflexive relation in A​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

THE TOTAL NUMBER OF SUBSETS OF A SET

If a set A contains n elements then number of subsets of A is

$\sf{ P(A) = {2}^{n} }$

THE TOTAL NUMBER OF REFLEXIVE RELATIONS ON A SET

If a set A contains n elements then the subsets of

A × A is

$\sf{ P(A \times \: A ) = {2}^{ {n}^{2} } }$

Now a reflexive relation on a set A is a subset of A × A that contains all elements of the set

$\sf{ \{(a \: ,\: a): a \in \: A\: \}}$

So the total number of reflexive relations are

$\sf{ \large{ {}^{ {n}^{2} - n} C_0}\: \: + \large{ {}^{ {n}^{2} - n} C_1}\: + \large{ {}^{ {n}^{2} - n} C_2} + .... + \large{ {}^{ {n}^{2} - n} C_{ {n}^{2} - n}} }$

$= \displaystyle \: \sf{ \: {2}^{( {n}^{2} - n)} \: }$

GIVEN

$\sf{A= \{1,2,3 \}}$

TO DETERMINE

The number of reflexive relation in A

CALCULATION

$\sf{ \:Here \: \: n (A) = 3 \: }$

Hence the total number of reflexive relations on the set A are

$= \displaystyle \: \sf{ \: {2}^{( {3}^{2} - 3)} \: }$

$= \displaystyle \: \sf{ \: {2}^{(9 - 3)} \: }$

$= \displaystyle \: \sf{ \: {2}^{6} \: }$

$= 64$

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Represent all possible one-one functions from the set A = {1, 2} to the set B = {3,4,5) using arrow diagram.

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