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  1. #1
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    Show that if A and B are sets then...

    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.
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  2. #2
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    Quote Originally Posted by tyrone92 View Post
    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?
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  3. #3
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    Quote Originally Posted by tyrone92 View Post
    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.
    $\displaystyle (A\cap B)\cup(A\cap\overline{B})=A$ simply says the any element in A is either in B or not in B.
    Can you write that up?
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