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The number of binary relations on a set with $n$ elements is:

1. $n^2$

2. $2^n$

3. $2^{n^2}$

4. None of the above

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In a binary relation two elements are chosen from the set. So, with $n$ elements $n^2$ pairings are possible. Now, a relation can be any subset of these $n^2$ pairings and thus we get $2^{n^2}$ binary relations.
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