Question

If n(A) =2 , n(B) =n and the number of relations from A to B is 256 then the value of n is​

Answer 1

Answer: n=4

Given:

$\sf n(A)=2$

$\sf n(B)=n$

$\sf Number\:of\:relations=256$

To find:

$\sf The\:value\:of\:n.$

Formula used:

$\boxed{\sf Number\:of\:relations=2^{m\times n}}$

$\sf where, m=Number\:of\:elements\:in\:set\:A$

          $\sf n=Number\:of\:elements\:in\:set\:B$

Explanation:

$\sf According\:to\:the\:question,$

$\sf n(A)=3\:and\:n(B)=n$

$\therefore \sf 256=2^{2\times n}$

$\Rightarrow \sf 256=2^{2n}$

$\Rightarrow \sf 2^8=2^{2n}$

$\sf On\:comparing\:powers,we\:get:$

$\Rightarrow \sf 8=2n$

$\Rightarrow \sf \dfrac{8}{2} =n$

$\Rightarrow\boxed{ \sf n=4}$

Therefore, the value of n is 4. That is, the number of elements in set B is 4.

Answer 2

Given ,

n(A) = 2

n(B) = n

n(A × B) = 256

We know that , The number of relation from x to y given by

$\boxed{ \sf{n(x \times y) = {2}^{ \{n(x) \times n(y) \}} }}$

Thus ,

$\sf \mapsto 256 = {2}^{(2 \times n)} \\ \\ \sf \mapsto {2}^{8} = {2}^{2n}$

On comparing the power , we get

8 = 2n

n = 8/2

n = 4

Therefore ,

The value of n is 4