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Every binary operation defined on a set having exactly one element is both commutative and associative.

I want to say it is true because the operation performed will result in the same assigned element no matter the order done in.

But, I want to say false because theres only one element and I can think of lots of examples where the set would not be closed under the binary operation.

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A binary operation $\star$ defined on the set $S$ is a function $S\times S\mapsto S$, so it is closed over $S$ by definition. The idea of closure only makes sense when talking about proper subsets of $S$.

The answer to the question is yes. Suppose $\star$ is a binary operation on $\{x\}$. Then if $a,b,c\in\{x\}$ we have $ab=ba$ and $a(bc)=(ab)c$, since everything is $x$.

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  • $\begingroup$ Is it fair to say every binary operation on this element is equality? $\endgroup$
    – Robert Frost
    Commented Jun 7, 2018 at 11:21

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