Question #4349

23 Let A and B be subsets of a universal set U. Prove the following.
(a). A\B = (U\B) \(U\A)

Expert's answer

1. Let x be from A\B. Therefore, x belongs A and x doesn't belong B.

As x belongs to A, x doesn't belong to U\A.

As x doesn't belong B, x belongs to U\B.

x doesn't belong to U\A and x belongs to U\B, so, x belongs to (U\A)\(U\B).

So, A\B belongs to (U\B)\(U\A).

2.Let x be from (U\B)\(U\A). Therefore, x belongs U\B and x doesn't belong U\A.

As x belongs to U\B, x doesn't belong to B.

As x doesn't belong U\A, x belongs to A.

x doesn't belong to B and x belongs to A, so x belongs to A\B.

So, (U\B)\(U\A) belongs to A\B.

A\B belongs to (U\B)\(U\A) and (U\B)\(U\A) belongs to A\B, so, A\B = (U\B)\(U\A).

As x belongs to A, x doesn't belong to U\A.

As x doesn't belong B, x belongs to U\B.

x doesn't belong to U\A and x belongs to U\B, so, x belongs to (U\A)\(U\B).

So, A\B belongs to (U\B)\(U\A).

2.Let x be from (U\B)\(U\A). Therefore, x belongs U\B and x doesn't belong U\A.

As x belongs to U\B, x doesn't belong to B.

As x doesn't belong U\A, x belongs to A.

x doesn't belong to B and x belongs to A, so x belongs to A\B.

So, (U\B)\(U\A) belongs to A\B.

A\B belongs to (U\B)\(U\A) and (U\B)\(U\A) belongs to A\B, so, A\B = (U\B)\(U\A).

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