Question

Assertion :$A=$ set of even natural numbers $<8$,$B=$ set of prime numbers $<7$ then numbers of relations from $A$ to $B$ are $2^6$ Reason: Number of relations from the set $A$ to $B$ are $2^{n\left(A\right)\times n\left(B\right)}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

Here, $A=\{2,4,6\}$

$B=\{2,3,5\}$

$\therefore n\left(A\right)=3,n\left(B\right)=3$

$A\times B=\left\{\right(2,2),(2,3),(2,5),(4,2),(4,3),(4,5),(6,2),(6,3),(6,5\left)\right\}$

$n(A\times B)=9$

So, the number of subsets of $A\times B$ is $2^9$

Since the number of relations from $A$ to $B$ is the number of subsets of $A\times B$.

So, the number of relation from $A$ to $B$ is $2^9$

Hence, the assertion is false.
Now, number of subsets of $A\times B=2^{n\left(A\right)\times n\left(B\right)}$

$=2^{n\left(A\right)\times n\left(B\right)}=2^{3\times 3}=2^9$

$\Rightarrow$ Assertion $\left(A\right)$ is false but Reason $\left(R\right)$ is true.