Are there any useful identities for simplifying an expression of the form:(adsbygoogle = window.adsbygoogle || []).push({});

$$((\ldots((x_{1} *_{1} x_{2}) *_{2} x_{3}) \ldots) *_{n - 1} x_{n})$$

Where each $$*_{i}$$ is one of $$\cap, \cup$$ and $$x_1 \ldots x_n$$ are sets?

I believe I found two; though I haven't proved them, I think they make sense:

$$((\ldots((x_{1} \cup x_{2}) *_{2} x_{3}) \ldots) \cup x_{1}) \equiv ((\ldots(x_{2} *_{2} x_{3}) \ldots) \cup x_{1})$$

$$((\ldots((x_{1} \cap x_{2}) *_{2} x_{3}) \ldots) \cap x_{1}) \equiv ((\ldots(x_{2} *_{2} x_{3}) \ldots) \cap x_{1})$$

Generally, how would you prove these? Just induction?

I tested a few expressions with the attached perl script which is why I think they work.

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# Identities of nested set algebraic expressions

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