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Math Help - A intersect (B U C)

  1. #1
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    A intersect (B U C)

    I need to prove A intersect (B U C)=(A intersect B)U(A intersect C).

    I'm having trouble getting started....
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    Quote Originally Posted by kathrynmath View Post
    I need to prove A intersect (B U C)=(A intersect B)U(A intersect C).

    I'm having trouble getting started....
    here's a start, prove that A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C) and then prove that (A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C), then you are done, since

    (X \subseteq Y) \wedge (Y \subseteq X) \implies X = Y

    i assume you know how to prove one set is a subset of another...am i wrong?
    Last edited by Jhevon; November 5th 2008 at 08:02 PM.
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  3. #3
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    Quote Originally Posted by kathrynmath View Post
    I need to prove A intersect (B U C)=(A intersect B)U(A intersect C).

    I'm having trouble getting started....

    To prove that:

    A \cap(BUC) = (A \cap B)U( A \cap C).

    We must prove that xεA \cap(BUC) implies xε(A \cap B)U( A \cap C).,and conversely.


    So start with that and see where this is going to get you.
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    Quote Originally Posted by Jhevon View Post
    here's a start, prove that A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C) and then prove that (A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C), then you are done, since

    (x \subseteq Y) \vee (Y \subseteq X) \implies X = Y

    i assume you know how to prove one set is a subset of another...am i wrong?
    So, can I let x be an element of A(BUC). So x is an elemnt of A and B or C?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kathrynmath View Post
    So, can I let x be an element of A(BUC). So x is an element of A and x is an element of (B or C)?
    yes, that is how you begin, you want to end up with x \in (A \cap B) \cup (A \cap C). that's one direction. then do it in the other and you're done
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    Quote Originally Posted by kathrynmath View Post
    So, can I let x be an element of A(BUC). So x is an elemnt of A and B or C?

    Let xεA mean x is an element of A e.t.c,et.c.


    Now xe[Α \cap ( BUC)] is equivalent to xεΑ and (xεB or xεC) due to the definition of intersection and union of sets.


    This is now the crucial point of the problem.Usually here a lot of people get stuck.Only those with a good knowledge of propositional logic have no problem.


    So put now :.........xεΑ =p,................xεB=q,................xεC=r,... .........and xεΑ and (xεB or xεC) becomes p^(qvr)........................................... .....................1


    (1) is now a statement in propositional logic where p,q,r can be any true or false proposition:

    So put :......p= i go to town ,.........q= i go to the movies,...........................r= i go to the theater and (1) becomes :




    i go to town and i go to the movies or to the theater:


    Now ask your logic what this is equivalent with??????????


    Once you come with an answer convert your answer into letters p,q,r using the above transformation.

    Finally put now p=xεA ....e.t.c,e.t.c.


    What is the result now?????
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