Simple tasks set theory

Discussion in 'Advanced Applied Math' started by marsimo777, Oct 16, 2021.

  1. marsimo777

    marsimo777

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    I have to prove that those set theories are true but I dont have a clue how to do it.
    Can somebody help me pls I only have 4 hours left..
     

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    marsimo777, Oct 16, 2021
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  2. marsimo777

    HallsofIvy

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    Those are of the form "set A= set B" or (statement 1-2 (a)) "statement A implies statement B" .

    To prove "set A= set B" first show that any member of A is necessarily a member of B (so that A is a subset of B), then turn around and show that any member of B is a member of A (so that B is a subset of A).

    For example, the first one says the complement of (M union N) is equal to (complement of M) intersect (complement of N).

    I presume you know that the "complement" of a set, A, is the set of elements, in the "universal set" that are NOT in A, that the "union" of two sets is the set of all elements that are in either set, and that the "intersection" of two sets is the set of all elements that are in both. For example if the universal set is {a, b, c, d, e, f, g} then the complement of {a, d, f} is {b, c, e, g}, the union of {a, d, f} and {a, c, f, g} is {a, c, d, f, g}, and the intersection of {a, d, f} and {a, c, f, g} is {a, f}.

    To show that the complement of (M union N) is equal to (complement of M) intersect (complement of N), start
    Suppose x is in complement of (M union N). Then x is not in M union N which means x is in neither M nor N. So x is in complement of M and in complement of N which means it is in (complement of M) intersect (complement of N).

    That proves that (M union N) is a SUBSET of (complement of M) intersect (complement of N). Now we need to prove "the other way around" which we do in basically the same way.

    Suppose y is in (complement of M) intersect (complement of N). Then y must be in both (complement of M) and (complement of N). Since y is in the complement of M it is not in M. Since y in the complement of N, it is not in N. y is not in M union N so it is in the complement of M union N.
     
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    HallsofIvy, Nov 7, 2021
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  3. marsimo777

    HallsofIvy

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    "1.2 (a)" is the "odd" one in that it does not ask you to prove that two set are equal but rather that the statement "M is a subset of N" implies the statement "the complement of N is a subset of the complement of M". We prove a statement "A implies B" by showing that whenever A is true, B is also true.

    Here start by assuming that M is a subset of N. That means that every member of M is a member of N. Now, suppose x is in the complement of N. Then x is NOT a member of N so is not in M. Therefore x is in the complement of M and we are done.

    (Unlike "=", subset is not "two-way" so we only have to prove one direction.)
     
    HallsofIvy, Nov 7, 2021
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