# Thread: A intersect (B U C)

1. ## 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....

2. Originally Posted by kathrynmath
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?

3. Originally Posted by kathrynmath
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.

4. Originally Posted by Jhevon
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?

5. Originally Posted by kathrynmath
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

6. Originally Posted by kathrynmath
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: