# Set Theory Question: a ∩ b ⊆ a

1. Jan 20, 2016

### YoshiMoshi

1. The problem statement, all variables and given/known data

I'm reviewing my powerpoints from class and see the formula A ∩ B ⊆ A. Is this a correct formula? I interpret this as all elements of set A intersected with set B is a subset of set A. I don't think this is a true statement, is it? Sorry it's been a while since I have studied set theory, probably back in high school days or so. I don't see how it could be true because elements of B are not necessarily elements of just A alone.

2. Relevant equations

3. The attempt at a solution

2. Jan 20, 2016

### phinds

What does A-intersect-B MEAN to you?

3. Jan 20, 2016

### YoshiMoshi

To me A intersect B is something like this

visual representation where A intersect B is the whole entire circle with all space in A and B included

That's why I don't understand, if I use this meaning and visual aid then certainly the space in B is not a subset of A. I don't understand why the equation would be true using this visual aid.

4. Jan 20, 2016

### SammyS

Staff Emeritus
No. It looks like you're thinking of union, not intersection.

5. Jan 20, 2016

### YoshiMoshi

I see this
http://www.wolframalpha.com/input/?i=(a+intersection+b)+subset+a
I understand in the vendiagram that it is true. However in the case of the picture posted previously

were B is considered the all the space inside B excluding the space in A (a circle with a hole in it)
A is considered it's on space independent of the space in B
then A intersect B is all the space in B plus the hole filled in as a solid circle

Ahhh this picture is a bad representation I think it makes since now A intersect B is were the space in A "overlaps the space in B" making the equation true. That was never to clear to me. Sorry for tangent question.

I think its better thought of as truth densities were A intersect B is the truth density of A and B occurring at the same time making the truth density of A a subs set of the truth density of both A and B occurring.

I always wondered, why is Boolean logic similar to set theory representation? I remember studying De Morgan's law in both digital logic class and some linear algebra class.

Thanks for the help!

6. Jan 20, 2016

### HallsofIvy

No! "A intersect B" is, by definition, the set of all points that are both set A and set B. In this case, that is exactly set A, not set B.
You can prove that $A\cap B\subseteq A$ by "Let x be a point in $A\cap B$. Then x is in both A and B. In particular x is in A. Since x can be any element of $A\cap B$ any member of $A\cap B$ is a member of A, by definition of "subset", $A\cap B\subseteq A$.
I have no idea what "truth densities" are. Perhaps it is a translation problem. Boolean logic is "similar" to set theory because Boolean logic has 2 values, "true", and "false" while a point can be in or not in a given set.