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Math Help - Proving Sets

  1. #1
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    Proving Sets

    A and B are sets i have to show proof that the following are equivalent

    (i) A(is a subset of ) B
    (ii) A(intersect)B = A
    (iii) A(union) B = B


    *note with (i) the symbol used is the subset symbol with a line underneath and im not sure if that means A=B or not.

    Thanks
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  2. #2
    Super Member Bacterius's Avatar
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    Note that as (i) implies that A is a subset of B, then it follows that the intersection of B with A equals A, and it also follows that the union of A with B is B. The other way round : if the union of A with B is B, then A must be a subset of B, and it follows that the intersection of B with A equals A.

    To help understand this, draw a diagram with A = \{4, 5, 7\} and B = \{2, 3, 4, 5, 6, 7, 8, 9\}
    Once you will have understood why this works, you will be able to put it in mathematical terms.

    Here are some basic ideas that can help you, too :

    If A is a subset of B, it means that all the elements of A are contained in B, among other elements of B
    If A intersect B = A, it means that the only elements that are in A and in B are the elements of B (equivalent to : A is a subset of B)
    If A union B = B, it means that the elements that are in A or in B, are equal to the elements of B (equivalent to : A is a subset of B)
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  3. #3
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    thank you.
    I understand the relationships between the different sets, however I am unsure as to how it should be represented as a proof.
    Using what you said I can write:

    (i) implies that A is a subset of B and if x (is in) A then x (is in) B

    A(intersection) B = A implies that
    if x (is in) A and x (is in) B then x (is in) B = A(is a subset of) B

    A(union)B = B implies that
    if x (is in) A or x (is in) B then x (is in) B = A(is a subset of) B

    therefore (i), (ii), (iii) are equivalent

    is this a proof or not?
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  4. #4
    Super Member Bacterius's Avatar
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    Sketched proof :

    (1) If A is a subset of B, then A intersect B = A
    (2) If A is a subset of B, then A union B = B
    (3) Therefore, by transitivity, if A is a subset of B then A intersect B = A and A union B = B.

    You only need to prove the weaker statements (1) and (2). Here is an example for (1) :

    Statement : If A is a subset of B, then A intersect B = A

    Proof : A is a subset of B, therefore each element of A is in B. This also implies that there exists no element of A which is not in B, and it follows that if A is a subset of B, then A intersect B = A.

    (2) can be proven in a similar way, and once you will have proven it you will have proven your original problem.
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  5. #5
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    how is this for proving (2):

    A is a subset of B

    therefore if x is in A then x is in B

    this also implies that if x is in A or x is in B then x is in B.

    and it follows that if A is a subset of B then A union B = B

    the proof sounds a bit off to me
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  6. #6
    Super Member Bacterius's Avatar
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    Yes, it is correct. Check out this one.

    Statement : If A is a subset of B, then A union B = B

    Proof : A union B returns the set of elements that are in A or in B. Say that A is a subset of B. It follows that any element that is in A is in B, and that the elements of B that are not in A are only in B. It immediately follows that A union B = B.

    Of course you can do it using letters, it equally works.
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  7. #7
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    thank you for your help
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