1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Set Theory Question: a ∩ b ⊆ a

  1. Jan 20, 2016 #1
    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. jcsd
  3. Jan 20, 2016 #2


    User Avatar
    Gold Member

    What does A-intersect-B MEAN to you?
  4. Jan 20, 2016 #3
    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.
  5. Jan 20, 2016 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    No. It looks like you're thinking of union, not intersection.
  6. Jan 20, 2016 #5
    I see this
    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!
  7. Jan 20, 2016 #6


    User Avatar
    Science Advisor

    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 [itex]A\cap B\subseteq A[/itex] by "Let x be a point in [itex]A\cap B[/itex]. Then x is in both A and B. In particular x is in A. Since x can be any element of [itex]A\cap B[/itex] any member of [itex]A\cap B[/itex] is a member of A, by definition of "subset", [itex]A\cap B\subseteq A[/itex].
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted