Thread: 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....

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 and then prove that , then you are done, since



i assume you know how to prove one set is a subset of another...am i wrong?

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

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


So start with that and see where this is going to get you.

Originally Posted by Jhevon

here's a start, prove that and then prove that , then you are done, since



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?

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 . that's one direction. then do it in the other and you're done

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 ( 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?????