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Math Help - Set Theory - empty set

  1. #1
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    Set Theory - empty set

    Just starting set theory (privately) and already a difficulty. A is a subset of B if all the elements of A are also in B. The empty set has no elements. But the empty set is a subset of A?? Seems illogical to me or have I missed something?
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  2. #2
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    Quote Originally Posted by wimpy0 View Post
    Just starting set theory (privately) and already a difficulty. A is a subset of B if all the elements of A are also in B. The empty set has no elements. But the empty set is a subset of A?? Seems illogical to me or have I missed something?
    A is a subset of B iff for all elements a of A a is in B.

    (Assume there exists a in the empty set, now a false proposition implies anything
    so this assumption implies that a is in B, so for all a in the null set a is in B)

    Also A is not a subset of B if there exists an element a of A which is not in B.

    Now this is always true of the empty set as there are no elements of A which are not in B.

    So the empty set is a subset of every set.

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    A is a subset of B iff for all elements a of A a is in B.

    (Assume there exists a in the empty set, now a false proposition implies anything
    so this assumption implies that a is in B, so for all a in the null set a is in B)

    Also A is not a subset of B if there exists an element a of A which is not in B.

    Now this is always true of the empty set as there are no elements of A which are not in B. (wimpy0 underlining)
    So the empty set is a subset of every set.

    RonL
    I read the line in your post which I've underlined above as " Now this is always true of the empty set as there are no elements of the empty set that are not in A".
    A true statement since the empty set has no elements .. period. This same argument would apply to all sets.

    Thanks. I think I've got it now (although I must admit that the initial clause before the "Also A" about a false proposition implying anything, is still making me wonder about its significance.
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