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This two well-known identity are very similar...

$$(a-b)^2=a^2+b^2-2ab$$ $$|A\Delta B|=|A|+|B|-2|A\cap B|$$

I know you can consider the operation $\Delta$ just like as sum modulo 2 and intersection as product. And it seems squaring is somehow like inner product. But i couldn't exactly derive them from each other. Can someone give me an exact proof that show why there are similar? Has some other algebraic identity any analogy in the set theory literature?

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If $a(x)$ and $b(x)$ are the indicator functions for sets $A$ and $B$ respectively ($a(x) = 1$ when $x \in A$, $0$ when $x \notin A$), then

$$ \eqalign{|A \Delta B| &= \sum_x (a(x)-b(x))^2\cr &= \sum_x \left(a(x)^2 + b(x)^2 - 2 a(x) b(x)\right)\cr &= \sum_x a(x) + \sum_x b(x) - 2 \sum_x a(x) b(x) \cr &= |A| + |B| - 2 |A\cap B|}$$

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    Is there any proof with respect to correspondence between operators of these spaces? I want to see some isomorphism... – 2017-01-11