I have no idea how to do (very very basic) proofs help guide me?

[nachosamurai]

Homework Statement

Let A and B be any sets.

1: Prove A is the disjoint union of A\B and A intersect B.

2: Prove A U B is the disjoint union of A\B, A intersect B, and B\A.

Homework Equations

?

The Attempt at a Solution

I understand most of the basic terminology used. I know disjoint means that no elements in one are in the other; on a Venn diagram they would not be overlapping at all. I know a union includes all elements from either set. I'm guessing a disjoint union just means that they are two disjoint sets?

I understand that these would all be disjoint, by use of Venn diagrams that I drew myself to help get my head around this new, alien type of math problem.

What I don't get is how to even begin proving this type of thing. My professor gave us some examples of other proofs, which often begin by supposing X is an element of one of the sets, and working with it. They were on different types of questions, though, so I can't figure out how to translate them to this particular question.

Honestly, I don't really understand how he comes up with his examples on any of this "discrete" math.

I'm not asking for you to do this problem for me, but rather to help me learn how to think logically and attack this type of thing myself. How do I even begin to construct a basic proof?

Assistance would be gratefully appreciated.

 

[FirstYearGrad]

It may be fruitful to think of the equivalency A=B as $$(A\subset B) \cap (B\subset A)$$. That is, show for an element a in A, show that it lies also within B, and for an element b in B, show that it also lies within A. For disjointedness, there are probably many ways to show it, but I think that if I were to solve it I might attack it by contradiction.

 

[iomtt6076]

First I just want to make sure I know what you're asking: did you mean to write:

"Prove $$A\cup B=(A\setminus B)\cup (A\cap B)\cup (B\setminus A)$$?"

 

[nachosamurai]

OK, I had a slight mistake in the problem. There are two separate but similar problems, and I got confused about which one I was reading half-way through the problem.

I've updated them in the original post, and will repeat it here as well:

Let A and B be any sets.

1: Prove A is the disjoint union of A\B and A intersect B.

2: Prove A U B is the disjoint union of A\B, A intersect B, and B\A.

The way they are written is identical to that. As I understand it, the "and" in these problems does not equal "union". Am I correct?

 

[iomtt6076]

Alright, let's deal with #1:
"prove $$A=(A\setminus B)\cup (A\cap B)$$"

First, do you see that $$A\setminus B=A\cap B^\complement$$?

 

[nachosamurai]

Well, I'm still confused why we're turning "and" into the union symbol. Is that what we're supposed to do?

As far as A\B = A intersect B-compliment, I understand that. Everything in A that's not in B is equal to everything that's in both A and everything that's not in B, and also in A.

But but but... and = union?

 

[iomtt6076]

Well, I'm still confused why we're turning "and" into the union symbol. Is that what we're supposed to do?

As far as A\B = A intersect B-compliment, I understand that. Everything in A that's not in B is equal to everything that's in both A and everything that's not in B, and also in A.

But but but... and = union?

The statement "union of X and Y" means $$X\cup Y$$.

Sorry, I don't mean to be difficult, but going back to what you wrote

1: Prove A is the disjoint union of A\B and A $$\cup$$ B.

that says to me, prove $$A=(A\setminus B)\cup (A\cup B)$$, which isn't true in general. Are you sure it isn't

1: Prove A is the disjoint union of A\B and A $$\cap$$ B.

 

[nachosamurai]

ah, shoot, yes >.< you're correct, it should be this:

1: Prove A is the disjoint union of A\B and A intersect B.

That's what i get for trying to mess around with these symbols, I have no idea how to do it correctly on these boards and didn't even notice I had made a union sign instead of an intersect sign. Going to stick to text for now.

 

[iomtt6076]

Okay, we're on the same page now.

First a comment, when you asked if and=union, perhaps you were thinking of if x is in A and B, then $$x\in A\cap B$$. But this is different to the 'and' in "union of X and Y"

Back to the problem. So you realize $$A\setminus B=A\cap B^\complement$$. We want to show $$A=(A\setminus B)\cup (A\cap B)$$. We can write the right hand side as $$(A\cap B^\complement)\cup (A\cap B)$$. Does this remind you of some identity?

 

[nachosamurai]

Thank you very much, I think I see how I can proceed from here :)

So basically it's all a matter of re-writing these to be easier to prove, eh?

 

[iomtt6076]

So basically it's all a matter of re-writing these to be easier to prove, eh?

For these kind of problems, yes. It might also be helpful to draw out the Venn diagrams to visualize what's going on.