Set Theory Proof Question.

[zacblack]

Prove the following using the pick-a-point method.

(A&B) U (A-B) = A

I know that this can be proven by examining venn diagrams, just not sure how to go about doing the proof using the pick a point method. Thanks for any help.

 

[pka]

Prove the following using the pick-a-point method.
(A&B) U (A-B) = A

First rewrite the LHS: \(\displaystyle (A\cap B)\cup (A\cap B')(\).

Now if \(\displaystyle x\in A\) then \(\displaystyle x\in B\text{ or }x\notin B\).

 

[stapel]

Prove the following using the pick-a-point method.

(A&B) U (A-B) = A

I know that this can be proven by examining venn diagrams, just not sure how to go about doing the proof using the pick a point method. Thanks for any help.

I've never heard of the "pick a point" method. I'm going to guess that it's a new name for what is also called "element-chasing", wherein one picks a generic element from a set and shows (by "chasing" its properties) that it also belongs in another set.

Since this is an "equals", rather than a "subset (in only one direction)", you'll need to show that the right-hand side (RHS) is a subset of the left-hand side (LHS), and vice versa. Let's try LHS-subset-of-RHS:

Pick an element from the LHS. By definition of "union", that element is either a member of A-intersect-B or else a member of A-complement-B. If the former, then the element is a member of A-intersect-B, so it is, in particular, a member of A. If the latter, then the element is a member of A-complement-B, so it is, in particular, a member of A....

And so on.