Exercise 286Logic, sets and proofs › Sets and mapsL1Added 4 Nov 2026

Cartesian product with seven elements

#Cartesian product#cardinality#impossible

Statement

Is there a set $A$ such that $A\times A$ has exactly $7$ elements?

Hint

If $|A|=n$ , then $|A\times A|=n^2$ . Is $7$ a perfect square?

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Setup. If $A$ has $n$ elements, then $A\times A$ has $n^2$ elements.

Check. $n^2=7$ has no integer solution since $2^2=4<7<9=3^2$ .

Conclusion. No such set exists.