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Proof about subsets.

  1. Jun 17, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that if A=B if and only if [itex] A \subseteq B [/itex] and [itex] B \subseteq A [/itex]
    3. The attempt at a solution

    If A is a subset of B then all the elements of A are in B . And if B is a subset of A then all the elements of B are in A . There fore there is a one-to-one correspondence between the 2 sets therefore they are equal.


    Is this to simple or do I need to be more rigorous.
     
  2. jcsd
  3. Jun 17, 2011 #2

    gb7nash

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    You might want to have a little more rigor. Do an element-wise proof.

    e.g.

    (<-) Fix x in A. Since A is a subset of B, ...

    Similarly, ...

    So, ...

    This only proves the backwards direction.
     
  4. Jun 17, 2011 #3

    Deveno

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    you aren't looking for ANY old 1-1 correspondence, you're looking for EQUALITY.

    if x is in A, and A ⊆ B, then x is in B.

    so B has every element A does (maybe more).

    but since B ⊆ A, there cannot be any element of B that is not in A

    (if there was, we quickly get a contradiction).

    so B has exactly the same elements as A, thus A = B.

    now, if A = B, why is A ⊆ B true?
     
  5. Jun 17, 2011 #4
    A subset can be equal to the set itself .
     
  6. Jun 18, 2011 #5

    vela

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    What's the definition of set equality?
     
  7. Jun 18, 2011 #6
    they have the same numbers of elements or the same cardinality.
     
  8. Jun 18, 2011 #7

    vela

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    So {1,2,3} and {3,4,5} are equal?
     
  9. Jun 18, 2011 #8
    ok so they have to have the same elements
     
  10. Jun 18, 2011 #9

    vela

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    So what specifically do you have to show to prove that two sets have the same elements?
     
  11. Jun 18, 2011 #10
    thanks for you help by the way . I have to show everything in A is in B and everything in B is in A . Ill work on it .
     
  12. Jun 18, 2011 #11

    vela

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    You just keep saying the same thing with different words, but there's a precise way to say "everything in A is in B" and vice versa. That's one thing you'll need to know to write a proper proof. So start by looking up what the precise definition of set equality is.
     
  13. Jun 18, 2011 #12

    HallsofIvy

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    Which is exactly the same thing as saying [itex]A\subseteq B[/itex].

    Which is exactly the same thing as saying [itex]B\subseteq A[/itex].

     
  14. Jun 20, 2011 #13
    if [itex] x \in A [/itex] then [itex] x \in B [/itex]
    if [itex] x \in B [/itex] then [itex] x \in A [/itex]
    therefore A=B
    It seems so simple. But Im not sure if that's good enough.
    Or am I going in circles .
     
  15. Jun 21, 2011 #14

    vela

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    It's not complete. When you have an if and only if, you have to prove both directions. In this case, you need to prove:

    (1) If A⊂B and B⊂A, then A=B.
    (2) If A=B, then A⊂B and B⊂A.

    You've essentially shown (1). Part of your confusion probably comes from thinking you're not really showing anything; that is, it's too obvious. With these really elementary proofs, you want the mindset that you can take nothing for granted and need to justify every little step, no matter how insignificant it may seem. If I were you, I'd add the step in red below to explicitly explain how you can conclude A=B.

    Assume A⊂B and B⊂A. By definition,

    A⊂B means (x∈A → x∈B)
    B⊂A means (x∈B → x∈A)

    Hence, we have x∈A ⟺ x∈B. Therefore, by definition of set equality, we can conclude A=B.

    [I'm assuming your definition of set equality is: A=B iff (x∈A ⟺ x∈B).]
     
  16. Jun 21, 2011 #15
    ok I see , Thanks you for our help.
     
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