Show that if A and B are sets then...

**tyrone92** · October 12th 2009, 11:15 AM

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.

**artvandalay11** · October 12th 2009, 12:02 PM

Originally Posted by tyrone92

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.

$(A\cap B) \cup (A\cap \overline{B})=A$

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 $x\in (A\cap B) \cup (A\cap \overline{B})$

Then by definition of union, $x\in (A\cap B)$ or $x\in (A\cap\overline{B})$

If $x\in (A\cap B)$ then $x\in A$ and $x\in B$ by definition of intersection. Therefore $x\in A$

If $x\in (A\cap\overline{B})$ then $x\in A$ and $x\in\overline{B}$ by definition of intersection. Therefore $x\in A$

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?

**Plato** · October 12th 2009, 12:06 PM

Originally Posted by tyrone92

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.

$(A\cap B)\cup(A\cap\overline{B})=A$ simply says the any element in A is either in B or not in B.
Can you write that up?

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