First slide

Introduction to sets

Question

Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second set. The values of m and n are, respectively,

Moderate

A

4,7
B

7,4
C

4,4
D

7,7

Solution

According to the question, we have
$\begin{array}{l} 2^{m} - 2^{n} = 112 \\ \Rightarrow 2^{n} (2^{m - n} - 1) = 2^{4} \times 7 \\ \Rightarrow 2^{n} = 2^{4} and 2^{m - n} - 1 = 7 \\ \Rightarrow n = 4 and 2^{m - n} = 8 \\ \Rightarrow n = 4 and 2^{m - n} = 2^{3} \\ \Rightarrow n = 4 and m - n = 3 \\ \Rightarrow n = 4 and m - 4 = 3 \\ \Rightarrow n = 4 and m = 7 \end{array}$

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