I am currently trying to prove the following:(adsbygoogle = window.adsbygoogle || []).push({});

An equation in X with righthand member [tex]\oslash[/tex] can be reduced to one of the form (A [tex]\cap[/tex] X) [tex]\cup[/tex] (B [tex]\cap[/tex] ~X) = [tex]\oslash[/tex].

(Where A, B, and X are sets of some universal set U, and ~X is the complement of the set X).

The only problem is that I'm not sure how to formulate or symbolize every possible equation in X. After asking a few friends and doing a bit of research online I came across ideas like structural induction and normal forms, but I'm still not sure how to apply it to prove this statement.

I know that once I can formulate this I can apply the deMorgan laws until complements of individual sets appear, and then expand the resulting lefthand side by the distributive laws and then play around with the X's and ~X's until I get what I need. But again, this all depends on the initial problem I have.

Any help would be appreciated.

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# Proof regarding the algebra of sets.

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