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Math Help - Prove using set algebra

  1. #1
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    Prove using set algebra

    Prove using set algebra:

    A U B = (A ∩ B') U (A' ∩ B) U (A ∩ B)
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  2. #2
    Super Member Matt Westwood's Avatar
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    Re: Prove using set algebra

    Quote Originally Posted by mathman11 View Post
    Prove using set algebra:

    A U B = (A ∩ B') U (A' ∩ B) U (A ∩ B)
    This might help you get started:

    https://proofwiki.org/wiki/Set_Diffe...form_Partition
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  3. #3
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    Re: Prove using set algebra

    Hello, mathman11!

    Prove using set algebra: . A \cup B \;=\; (A \cap B') \cup (A' \cap B) \cup (A \cap B)

    There are two rules for which I do not know the names.

    I will call them: . \begin{Bmatrix}S \cup S' \:=\:U && \text{Property 1} \\ S \cap U \:=\:S && \text{Property 2} \end{Bmatrix}


    Consider the right side.

    \begin{array}{cccccc}1. & (A\cap B') \cup (A' \cap B) \cup (A \cap B) && 1. & \text{Given} \\ \\ 2. & (A\cap B') \cup \left[(A' \cap B) \cup (A \cap B)\right] && 2.& \text{Associative} \\ \\ 3. & (A \cap B') \cup\left[(A' \cup A) \cap B\right] && 3. & \text{Distributive} \\ \\ 4. & (A \cap B') \cup (U \cap B) && 4. & \text{Prop. 1} \\ \\ 5. & (A \cap B') \cup B && 5. & \text{Prop. 2} \\ \\ 6. & (A \cup B) \cap (B' \cup B) && 6. & \text{Distributive} \\ \\ 7. & (A \cup B) \cap U && 7. & \text{Prop. 1} \\ \\ 8. & A\cup B && 8. & \text{Prop. 2} \end{array}

    Thanks from mathman11 and topsquark
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    Re: Prove using set algebra

    thanks Soroban this was really helpful.

    Do you have any tips as to how to approach questions like the one you just did?
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