The equation $(x+x)+x=x+(x+x)$ is an instance of both the associative and commutative properties. Also, the equation $x+x=x+x$ is an instance of both the commutative property and the reflexive property of equality. That raises the question, is there an equation which is an instance of both the associative property and the reflexive property of equality?
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That is an interesting question! Wouldn’t that be the first equation $(x + x) + x = x + (x + x)$? – Air Mike Sep 04 '20 at 20:16
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1@Air: no, the two terms on the two sides are different so the reflexive property of equality doesn't (directly) apply. – Qiaochu Yuan Sep 04 '20 at 21:03
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Does $(x + 0) + x = x + (0 + x)$ work? Since $x + 0 = 0 + x = x$. – Sam Freedman Sep 04 '20 at 21:53
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@SamFreedman No, because the terms on both sides are distinct. – user107952 Sep 05 '20 at 01:31
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No. If an equation is an instance of associativity, it must have the form $$(s*t)*u = s*(t*u)$$ where $s$, $t$, and $u$ are terms. If that equation is an instance of reflexivity, the two sides must be identical, so $s$ and $s*t$ are identical, and $u$ and $t*u$ are identical. But no term can be identical to a proper subterm, so we have a contradiction.
Alex Kruckman
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