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!

Simple Set Theory - Union, Intersect and Complement

  1. Jul 24, 2011 #1
    1. The problem statement, all variables and given/known data
    If A, B, and C are subsets of the set S, show that
    [tex]A^C \cup B^C = \left(A \cap B\right)^C[/tex]


    2. Relevant equations
    [tex]A^C = \{x \in S: x \not \in A\}[/tex]
    [tex]A\cup B = \{x \in S:\; x \in A\; or\; x\in B\}[/tex]
    [tex]A\cap B = \{x \in S:\; x \in A\; and\; x\in B\}[/tex]


    3. The attempt at a solution

    [tex]A^C \cup B^C = \{x\in S:\; x\not \in A\; or\; x \not \in B\}[/tex]
    [tex]\left(A \cap B\right)^C = \{x\in S:\; x \not \in \left(A\cap B)\right \}[/tex]
    Since [itex]\lnot A \lor \lnot B = \lnot \left(A \land B\right)[/itex], we have that [itex]\{x\in S:\; x\not \in A\; or\; x \not \in B\} = \{x\in S:\; x \not \in \left(A\cap B)\right \}[/itex]
    [tex]\;\;\; \therefore A^C \cup B^C = \left(A \cap B\right)^C[/tex]

    Is it valid for me to relate the set operations to the analogous logic symbols to show that this is true? Such as, complement being the same as [itex]\lnot[/itex] (not) or the union and intersection being the same as [itex]\lor[/itex] (or) and [itex]\land[/itex] (and), respectively?

    I'm sorry if this is obvious. I'm teaching myself Analysis out of Maxwell Rosenlicht's "Introduction to Analysis" text, so I want to make sure I'm not making any false assumptions or developing a false intuition of set theory.
     
  2. jcsd
  3. Jul 24, 2011 #2
    It's okay to use "not" when dealing with the complement. It's also okay to use the equivalence of one thing to another, like -A or -B = -(A and B) as long as you know it's true, and you are sure to handle order of operations in these situations properly. The only time I can think of where it may be not okay to use such equivalences would be on a test, where the whole point would be showing the equivalence.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Simple Set Theory - Union, Intersect and Complement
Loading...