First slide

Introduction to sets

Question

Two finite sets have m and n elements. The total number of subsets of the first set is 48 more than the total number of subsets of the second set. The respective values of m and n are

Easy

A

7,6
B

6,3
C

6,4
D

7,4

Solution

$\begin{array}{l} Since 2^{m} - 2^{n} = 48 = 16 \times 3 = 2^{4} \times 3 \\ \begin{aligned} \Rightarrow 2^{n} (2^{m - n} - 1) = 2^{4} \times 3 \\ 2^{n} = 2^{4}, 2^{m - n} - 1 = 3 \\ \Rightarrow n = 4 and 2^{m - n} = 4 = 2^{2} \end{aligned} \\ \Rightarrow m - n = 2 \\ \Rightarrow m - 4 = 2 \\ \Rightarrow m = 6 \\ \Rightarrow m = 6, n = 4 \end{array}$

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