First slide

Operations on Sets

Question

Two finite sets have m and n(m, n) elements. The number of subsets of the first set is 112 more than that of the second set. The value of mn is

Moderate

A

18
B

28
C

32
D

36

Solution

According to the question,
$\begin{aligned} 2^{m} - 2^{n} = 112 \\ \Rightarrow 2^{n} (2^{m - n} - 1) = 2^{4} \times 7 \end{aligned}$
Compering exponents on both side, we get
$2^{n} = 2^{4} and 2^{m - n} - 1 = 7$
Now, $2^{n} = 2^{4}$
$\Rightarrow n = 4$
And $2^{m - n} = 8$
$\begin{array}{ll} \Rightarrow 2^{m - n} = 2^{3} \\ \Rightarrow m - n = 3 \\ \Rightarrow m - 4 = 3 \\ ∴ m = 7 \end{array}$

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