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Homework Help: 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.
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