if A and B are sets, then (A n B ) U ( A n B ) = A
the 2nd B has a _ on the top. the n is supposed to be an upside down U
How would i go about showing this.
$\displaystyle (A\cap B) \cup (A\cap \overline{B})=A$
The general strategy for proving 2 sets are equal is to prove that all of the elements in the one on the left are also in the one of the right and viceversa
So, let $\displaystyle x\in (A\cap B) \cup (A\cap \overline{B})$
Then by definition of union, $\displaystyle x\in (A\cap B)$ or $\displaystyle x\in (A\cap\overline{B})$
If $\displaystyle x\in (A\cap B)$ then $\displaystyle x\in A$ and $\displaystyle x\in B$ by definition of intersection. Therefore $\displaystyle x\in A$
If $\displaystyle x\in (A\cap\overline{B})$ then $\displaystyle x\in A$ and $\displaystyle x\in\overline{B}$ by definition of intersection. Therefore $\displaystyle x\in A$
Now you should try going the other way. Does the question state B is subset of A? or a subset of X, to give the complement meaning?