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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- Nov 4th 2008, 06:33 AMkathrynmathA 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.... - Nov 4th 2008, 07:30 AMJhevon
here's a start, prove that $\displaystyle A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$ and then prove that $\displaystyle (A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C)$, then you are done, since

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

i assume you know how to prove one set is a subset of another...am i wrong? - Nov 4th 2008, 07:37 AMpoutsos.B

To prove that:

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

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

So start with that and see where this is going to get you. - Nov 4th 2008, 09:44 AMkathrynmath
So, can I let x be an element of Ahttp://www.mathhelpforum.com/math-he...c7f8f01c-1.gif(BUC). So x is an elemnt of A and B or C?

- Nov 5th 2008, 07:04 PMJhevon
- Nov 7th 2008, 02:21 AMpoutsos.B

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

Now xe[Α$\displaystyle \cap$ ( 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?????