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# Answer to Question #17880 in Discrete Mathematics for Maha

Question #17880
let A,B and C be sets. show that (A-B)-C=(A-C)-(B-C)
1
2012-11-06T09:35:09-0500
(A-C)-(B-C) = (A n C&#039;) n (B n C&#039;)&#039;
=(A n C&#039;) n (B&#039; u C&#039;&#039;)
=(A n C&#039;) n (B&#039; u C)
= A n (C&#039; n (B&#039; u C))
= A n ((C&#039; n B&#039;) u (C&#039; n C))
= A n ((C&#039; n B&#039;) u emptyset)
= A n (C&#039; n B&#039;)
= A n (B&#039; n C&#039;)
= (A n B&#039;) n C&#039;
= (A-B)-C

All we are using here is the definition of &#039;, identities like X&#039;&#039; = X, the associativity and commutativity of n, &quot;DeMorgan&#039;s law&quot; (X n Y)&#039; = X&#039; u Y&#039;, a few &#039;distributive laws&#039; like X n (Y u Z) = (X n Y) u (X n Z), and a property of the empty set.

If you read this &quot;backwards&quot; you get a proof that starts from the left hand side; if you got stuck going that way, it is probably because the &quot;step&quot; writing C&#039; n B&#039; as (C&#039; n B&#039;) union the empty set and then writing the empty set in a new way is not at all obvious. (If you had drawn a venn diagram or something representing the situation, it might make sense from that, but from algebra it is not at all clear.)

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Assignment Expert
09.11.12, 08:13

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Maha
09.11.12, 06:46

Thanks :)