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Math Help - Basic Set Theory Proofs

  1. #1
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    Basic Set Theory Proofs

    I'm taking Topology next semester and, to prepare, I'm learning a little set theory. I had bought a Topology book from Dover Publications a couple years ago and am now starting to do some problems in it. The very first section is on Set Theory. Here's the first couple problems:

    1. S ⊂ T, then T - (T - S) = S.

    Proof:
    Let x ∈ (S ⊂ T). Therefore, x ∉ (T S). If x ∉ (T S), then x ∈ T (T S). And x ∈ (S ⊂ T). Therefore, S ⊂ (T - (T - S))

    Let x ∈ (T - (T - S)). This implies that x ∉ (T S). Which implies that x ∈ S. Therefore, T (T S) ⊂ S and T - (T - S) = S.

    2. S ⊂ T iff S ∩ T = S.

    Proof:
    If S ⊂ T, every x ∈ S is in T. That implies that every x ∈ S is in S ∩ T. This implies that S ∩ T = S.

    Let x ∈ S. If S ∩ T = S, then x ∈ (S ∩ T). This implies that S is contained in T and S ⊂ T.

    I could really use some direction/correction.
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  2. #2
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    Re: Basic Set Theory Proofs

    First, the symbol ⊂ is somewhat ambiguous because it may mean either proper or improper subset in different sources. To clear the ambiguity, ether use ⊆ for improper subset and ⊊ for proper subset or describe your notation in words.

    Quote Originally Posted by aareavis View Post
    1. S ⊂ T, then T - (T - S) = S.

    Proof:
    Let x ∈ (S ⊂ T).
    An object x cannot belong to (S ⊂ T) because (S ⊂ T) is a proposition (something that is either true or false), not a set. It is sometimes possible to write x ∈ S ⊆ T as a contraction to "x ∈ S and S ⊆ T," but such shortcuts are better avoided in the beginning.

    Quote Originally Posted by aareavis View Post
    If x ∉ (T S), then x ∈ T (T S).
    Only if x ∈ T, which, granted, is apparently the case here.

    Quote Originally Posted by aareavis View Post
    Let x ∈ (T - (T - S)). This implies that x ∉ (T S). Which implies that x ∈ S.
    Or x ∉ T.

    Quote Originally Posted by aareavis View Post
    2. S ⊂ T iff S ∩ T = S.

    Proof:
    If S ⊂ T, every x ∈ S is in T. That implies that every x ∈ S is in S ∩ T. This implies that S ∩ T = S.
    Strictly speaking, it just implies that S ∩ T ⊇ S. The other inclusion is obvious, but at this high level of proof detail maybe it should be said explicitly.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Re: Basic Set Theory Proofs

    Quote Originally Posted by aareavis View Post
    1. S ⊂ T, then T - (T - S) = S.
    If you have already studied the concept of universal set U , the complementary M^c of a subset M\subset U, distributive and Morgan's laws, etc you can prove the equality in another way.

    Choosing as universal set any set U such that S\cup T\subset U (in this case for example U=T) we have M-N=M\cap N^c for M,N\subset U . So,

    T-(T-S)=T\cap (T-S)^c=T\cap (T\cap S^c)^c=T\cap (T^c\cup S)=

    (T\cap T^c)\cup(T\cap S)=\emptyset \cup (T\cap S)=T\cap S=S
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    Re: Basic Set Theory Proofs

    Quote Originally Posted by FernandoRevilla View Post
    If you have already studied the concept of universal set U , the complementary M^c of a subset M\subset U, distributive and Morgan's laws, etc you can prove the equality in another way.

    Choosing as universal set any set U such that S\cup T\subset U (in this case for example U=T) we have M-N=M\cap N^c for M,N\subset U . So,

    T-(T-S)=T\cap (T-S)^c=T\cap (T\cap S^c)^c=T\cap (T^c\cup S)=

    (T\cap T^c)\cup(T\cap S)=\emptyset \cup (T\cap S)=T\cap S=S
    I have studied the concept of a universal set (Prob & Stat). I just didn't think I could use it here. Now that I think about it, the way my book describes it is this: T - S is "the compliment of S in T" which is the same as saying T ∩ S. (I use apostrophe instead of C. Habit from Prob & Stat.)
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Re: Basic Set Theory Proofs

    Quote Originally Posted by aareavis View Post
    I have studied the concept of a universal set (Prob & Stat). I just didn't think I could use it here. Now that I think about it, the way my book describes it is this: T - S is "the compliment of S in T" which is the same as saying T ∩ S. (I use apostrophe instead of C. Habit from Prob & Stat.)
    Well, in that case you can use the alternative way I provided (if you have previously covered distributive laws, etc).
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