1. Not finding help here? Sign up for a free 30min 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!

Element proof for sets A,B,C

  1. Jul 16, 2014 #1
    1. The problem statement, all variables and given/known data

    Prove

    [tex]A \times (B \cap C) = (A \times B) \cap (A \times C) [/tex]

    3. The attempt at a solution

    Let [tex]x \in A[/tex] and [tex]y \in B \cap C \rightarrow y \in B \wedge y \in C[/tex]

    now [tex] \exists (x,y) \in A \times (B \cap C) [/tex]

    so [tex](x,y) \in A \times B \wedge (x,y) \in A \times C[/tex]

    thus [tex](x,y) \in (A \times B) \cap (A \times C) [/tex]

    therefore [tex]A \times (B \cap C) = (A \times B) \cap (A \times C) [/tex]
     
  2. jcsd
  3. Jul 17, 2014 #2

    verty

    User Avatar
    Homework Helper

    A,B,C could all be the empty set, in which case your third line is incorrect, there may not exist such (x,y).

    I would use set-builder notion to show that they are the same.
     
  4. Jul 17, 2014 #3
    If they are all the empty set then thats pretty trivial and not interesting. And all the same? As in A =B=C?
     
  5. Jul 17, 2014 #4

    verty

    User Avatar
    Homework Helper

    I mean the left and right-hand sides, you want to show that they are the same set.
     
  6. Jul 17, 2014 #5

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    I addition to what verty pointed out:

    You have only done half of the proof.

    You showed that the left hand side is a subset of the right hand side.
     
  7. Jul 17, 2014 #6

    Ah yes. I went back and showed it goes both ways. Thanks
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted