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 $\displaystyle A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$ 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?
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.
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?????