Question

Find the value of m if \[\text{n(A)=2}\], \[\text{n(B)}\,\text{=}\,\text{m}\] and the number of relation from A to B is 64.
(a) 6
(b) 3
(c) 16
(d) 8

Answer 1

Hint: We know that the number of different relations from A to B is \[{{2}^{\text{xy}}}\] and number of relation from A to B is mentioned in the question as 64 and also number of elements of set B is given. Hence we will use these inputs to find the value of m.

Complete step-by-step answer:
Before proceeding with the question we must understand the concept of sets and relations.
A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product \[\,\text{A }\!\!\times\!\!\text{ B}\]. The subset is derived by describing the relationship between the first element and the second element of the ordered pairs in \[\,\text{A }\!\!\times\!\!\text{ B}\].
If A has x elements and B has y elements, then \[\,\text{A }\!\!\times\!\!\text{ B}\] has \[\text{x }\!\!\times\!\!\text{ y}\] element. And the number of different relations from A to B is \[{{2}^{\text{xy}}}\].
Number of given elements in set A and set B is mentioned in the question, so using this information we get,
Number of elements in set A: \[\text{n(A)=2}.......\text{(1)}\]
Number of elements in set B: \[\text{n(B) = m}.......\text{(2)}\]
And the number of relations from A to B is given in the question as 64 and the formula for the number of different relations from A to B is \[{{2}^{\text{xy}}}\].
 \[\Rightarrow {{2}^{\text{xy}}}=64.......(3)\]
Here from equation (1) and equation (2) we get x as 2 and y as m and substituting these values in equation (3) we get,
 \[\Rightarrow {{2}^{2\text{m}}}=64.......(4)\]
We know that 2 to the power 6 is 64, so changing 64 in terms of powers of 2 we get,
 \[\Rightarrow {{2}^{2\text{m}}}={{2}^{6}}.......(5)\]
As the base is 2 on both sides of equation (5) we equate the powers and then solve for m we get,
\[\begin{align}
  & \,\Rightarrow 2\text{m}=6 \\
 & \,\Rightarrow \text{m}\,\text{=3} \\
\end{align}\]
Hence the number of elements of set B is 3. The correct answer is option (b).

Note: Remembering the formula of the number of relation from A to B is the key here. We can commit a mistake in solving equation (4) if we do not know that 2 to the power 6 is 64 and hence we will not be able to proceed further.