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Thread: Proof by cases to prove distributive law for sets

  1. #1
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    Proof by cases to prove distributive law for sets

    https://gyazo.com/defe1638bbbf385f343684bd6c024c3c

    So I know what proof by cases is... and I know how to prove problems like If n^2 is even, then n is even.

    I got lost when the problems involve sets and intersection/unions..

    Could someone explain this to me Barney-style?
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Proof by cases to prove distributive law for sets

    Quote Originally Posted by MrJank View Post
    https://gyazo.com/defe1638bbbf385f343684bd6c024c3c

    So I know what proof by cases is... and I know how to prove problems like If n^2 is even, then n is even.

    I got lost when the problems involve sets and intersection/unions..

    Could someone explain this to me Barney-style?
    I think what they are after is how the sets "interact" with each other. For example, start with all of A, B, C disjoint from each other. Then move to the case where A and B intersect but C is disjoint from both. Then where A and C intersect and B is disjoint, etc. etc.

    -Dan
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    Re: Proof by cases to prove distributive law for sets

    Quote Originally Posted by MrJank View Post
    https://gyazo.com/defe1638bbbf385f343684bd6c024c3c
    So I know what proof by cases is... and I know how to prove problems like If n^2 is even, then n is even.
    I got lost when the problems involve sets and intersection/unions..
    Could someone explain this to me Barney-style?
    Why don't you learn to post symbols? Many here will not click on a link.
    It is really easy: \$A\cap(B\cup C)=(A\cap B)\cup(A\cap C)\$ gives $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$
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  4. #4
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    Re: Proof by cases to prove distributive law for sets

    The problem is to prove that "$\displaystyle A\cap (B\cup C)= (A\cap B)\cup (A\cap C)$

    To prove that "set X= set Y" do two "cases"
    1) show that "if p is in X then p is in Y".
    2) show that "if p is in Y then p is in X".

    1) "If p is in $\displaystyle A\cap(B\cup C)$ then p is in $\displaystyle (A\cap B)\cup (A\cap C)$"
    If p is in $\displaystyle A\cap (B\cup C)$ then p is in A and in $\displaystyle B\cup C$
    Since p is in $\displaystyle B\cup C$ then either p is in B or p is in C so we have two "subcases":
    1a) p is in B. Then since p is in A and in B p is in $\displaystyle A\cap B$ and so in $\displaystyle (A\cap B)\cup (A\cap C)$.
    1b) p is in C. Then since p is in A and in C p is in $\displaystyle A\cap C$ and so in $\displaystyle (A\cap B)\cup (A\cap C)$.

    2) "If p is in $\displaystyle (A\cap B)\cup (A\cap C)$ then p is in $\displaystyle A\cap(B\cup C)$".
    Since p is in $\displaystyle (A\cap B)\cup (A\cap C)$ p is either in $\displaystyle A\cap B$ or p is in $\displaystyle A\cap C$
    so again we have two "subcases":
    2a) if p is in $\displaystyle A\cap B$ then p is in both A and B. Since p is in B, b is in $\displaystyle B\cup C$ so p is in $\displaystyle A\cap(B\cup C)$.
    2b) if p is in $\displaystyle A\cap C$ then p is in both A and C. Since p is in C, b is in $\displaystyle B\cup C$ so p is in $\displaystyle A\cap(B\cup C)$.
    Thanks from topsquark and MrJank
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  5. #5
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    Re: Proof by cases to prove distributive law for sets

    Quote Originally Posted by MrJank View Post
    I got lost when the problems involve sets and intersection/unions..
    Could someone explain this to me Barney-style?
    In thirty-five years of teaching this course, I have never heard of Barney-style.
    If $x\in A\cap(B\cup C)$ means that $x\in A$ AND $x\in B\cup C$
    $x\in B\cup C$ means that $x\in B$ OR $x\in C$
    Putting those two together we get:
    $x\in A~\&~x\in B$ OR $x\in A~\&~x\in C$
    Thus $x\in (A\cap B)\cup(A\cap C)$

    At least you can show some effort to do the reverse.
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  6. #6
    Forum Admin topsquark's Avatar
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    Re: Proof by cases to prove distributive law for sets

    Quote Originally Posted by Plato View Post
    In thirty-five years of teaching this course, I have never heard of Barney-style.
    I hate to say I recognize the reference but he's referring to "Barney the Purple Dinosaur." In other words, in simple terms.

    -Dan
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    Re: Proof by cases to prove distributive law for sets

    I thought maybe "Deputy Barney" of the Andy Griffeth Show (In reruns called "Andy of Mayberry").
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