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proofs: set theory problem

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Oct 22nd, 2017

Denote AU(B∩C) as D1 and (AUB)∩C as D2. We know also that AcC.

This means that AUC = C (the greater set) and A∩C = A (the less set).

Then D1 = AU(B∩C) = (AUB)∩(AUC) = (AUB)∩C = D2.

Or D2 = (AUB)∩C = (A∩C)U(B∩C) = AU(B∩C) = D1. That's it!

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Or less formally: let's prove for any xϵU xϵD1 implies xϵD2, and vice versa. This will mean that the sets D1 and D2 are equal.

1. Prove "xϵD1 implies xϵD2".
If xϵD1 then xϵA or (xϵB and xϵC).
In the first case (xϵA), xϵ(AUB) and xϵC because of AcC. So, xϵ(AUB)∩C = D2.
In the second case, xϵB and xϵC. Then xϵ(AUB) and xϵC. So, xϵ(AUB)∩C = D2.

2. Prove "xϵD2 implies xϵD1".
If xϵD2 then xϵC and (xϵA or xϵB), it is equivalent to (xϵC and xϵA) or (xϵC and xϵB).
In the first case (xϵC and xϵA), xϵA and this is sufficient for xϵAU(B∩C)=D1.
In the second case, (xϵC and xϵB). Then xϵ(B∩C) and this is sufficient for xϵAU(B∩C)=D1.
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Apr 6th, 2015

thank you!

Apr 29th, 2015

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Oct 22nd, 2017
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Oct 22nd, 2017
Oct 22nd, 2017
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