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Math Help - Question about power sets and the empty set element

  1. #1
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    Question about power sets and the empty set element

    Say theres a set A = {1}, set B = {2}

    Then the powerset of A is P(A) = {{1}, Φ}, and powerset of B is P(B) = {{2}, Φ}

    Now if I have P(A) - P(B), then this wouldn't remove the empty set element would it?

    I know the empty set is a subset of all sets, and it doesn't really make sense that you could remove the empty set.

    So I'm guessing P(A) - P(B) = {{1}, Φ} still.

    Is this right? Or would P(A) - P(B) = {{1}}?

    Thanks for the help!
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  2. #2
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    I surmise you mean set difference. I.e., P(A) - P(B) = {x |x in P(A) but x not in P(B)}.

    In this case, the empty set (0) is NOT in P(A) - P(B).

    0 is in P(B) so it cannot be in P(A) - P(B).

    That 0 is a subset of every set doesn't entail that 0 is a MEMBER of every set. 0 is a subset of P(A) - P(B) but 0 is not a member of P(A) - P(B).

    You're right when you say P(A) - P(B) = {{1}}, since {1} is the only element of P(A) that is not an element of P(B).
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  3. #3
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    Thanks for your quick reply! Appreciate the help.
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  4. #4
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    Quote Originally Posted by swtdelicaterose View Post
    Say theres a set A = {1}, set B = {2}
    Then the powerset of A is P(A) = {{1}, Φ}, and powerset of B is P(B) = {{2}, Φ}
    Now if I have P(A) - P(B), then this wouldn't remove the empty set element would it? P(A) - P(B) = {{1}}?
    That is correct.
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