First slide

Relations XII

Question

Suppose A and B are two finite sets. If the number of relations that can be defined on A is 496 more than the number of number of
relations that can be defined on B, then the elements is A =

Difficult

Solution

Suppose the number of elements in A is n and the
number of elements in B is m. We have
$\begin{aligned} 2^{n^{2}} - 2^{m^{2}} = 496 \\ \Rightarrow & 2^{n^{2}} \geq 496 \Rightarrow 2^{n^{2}} \geq 512 = 2^{9} \\ ∴ & n = 3 \end{aligned}$
For n= 3, m = 2

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