Show that if A and B are sets, then (A n B) u (A n B) = A.

We prove that the left side is contained in the right side, and vica versa.

First, L.S. contained in the R.S.

Let x belong to the left hand side. Then x is in (A n B) or x is in (A n B_bar).

Suppose x is in (A n B). Then is x is in A, and x is in B. But if x is in A, that means x belongs to the right hand side.

Suppose x is in (A n B_bar). Then is x is in A, and x is in B_bar. Again, we conclude that x belongs to the right hand side.

Now we prove that the R.S. is contained in the L.S.

Let x be in R.S. That means x is in A. If x is in A then x is also in (A n B), which makes x an element of the union between (A n B) and (A n B_bar). So x is in the left side.

Hence the two sides are equal.

I hope this helps.