Question

Two finite sets have elements. The total number of subsets of the first set is 48 more than the total number of subsets of the second set. Find $m \text{ and } n$ .

Answer 1

In order to answer this question, to know the value of $m \text{ and } n$ with respect to the given question, first we will assume two variables which have $m \text{ and } n$ number of elements. Then we will follow the instructions given in the question itself.

Complete step-by-step solution:
Let A have $m$ elements.
Let B have $n$ elements.
Total number of subsets of A $= {2^m}$
Total number of subsets of B $= {2^n}$
According to the question, now we have an equation:-
${2^m} - {2^n} = 48$
Now, we will take common ${2^n}$ from the L.H.S-
$\Rightarrow {2^n}({2^{m - n}} - 2) = 48$
So, ${2^n} = \text{even}\,\text{ and }\,{2^{m - n}} - 2 = 0\,even$
Now,
$\begin{aligned} &48 = 8 \times 6 = {2^3} \times {6^1} \\\ & \Rightarrow {2^n}({2^{m - n}} - 2) = {2^3} \times 6 \\\ &\Rightarrow n = 3 \\\ \end{aligned}$
Now,
$\begin{aligned} &8({2^{m - 3}} - 2) = 6\times 8 \\\ &\Rightarrow {2^{m - 3}} - 2 = 6 \\\ & \Rightarrow {2^{m - 3}} = 8 \\\ & \Rightarrow {2^{m - 3}} = {2^3} \\\ &\Rightarrow m - 3 = 3 \\\ & \Rightarrow m = 6 \\\ \end{aligned}$
Therefore, the value of $m \text{ and } n$ are 6 and 3 respectively.

Note: A subset is a collection of elements that also appear in another collection. Remember that a set is a group of related elements. For example, $\\{ a,b,c,d\\}$ is a set of letters, while $\\{ cat,dog,fish,bird\\}$ is a set of animals. $\\{ 2,4,6,8,10\\}$ is a set of even numbers, and $\\{ a,b,c,d\\}$ is a set of letters.