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Math Help - Show that if A and B are sets with A ⊆ B, then A ∪ B = B.

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    Show that if A and B are sets with A ⊆ B, then A ∪ B = B.

    Show that if A and B are sets with A ⊆ B, then A ∪ B = B.

    Is A ∪ B = B. because A ∪ B, Means A or B or Both ?
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    Re: Show that if A and B are sets with A ⊆ B, then A ∪ B = B.

    Quote Originally Posted by lamentofking View Post
    Show that if A and B are sets with A ⊆ B, then A ∪ B = B.
    Is A ∪ B = B. because A ∪ B, Means A or B or Both ?
    Basically, yes.
    Realize, That if X is a set and Y is any other set then X\subseteq (X\cup Y).

    So in the case B\subseteq (A\cup B). Now all you need do is show A\cup B\subseteq B.
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    Re: Show that if A and B are sets with A ⊆ B, then A ∪ B = B.

    Quote Originally Posted by Plato View Post
    Now all you need do is show A\cup B\subseteq B.
    Is the answer because A U B means that A or B so it can be included in B ?
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    Re: Show that if A and B are sets with A ⊆ B, then A ∪ B = B.

    Quote Originally Posted by lamentofking View Post
    Is the answer because A U B means that A or B so it can be included in B ?
    Yes.
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    Re: Show that if A and B are sets with A ⊆ B, then A ∪ B = B.

    The rigorous way of showing that " X= Y" is to show that " X\subseteq Y" and that " Y\subseteq X". And to show that " X\subseteq Y" start with "if x\in X and use the conditions on X and Y to conclude "therefore x\in Y".

    So to show that, A\cup B= B you must first show A\cup B\subseteq B. And you do that by saying "if x\in A\cup B then either x\in A or x\in B (by definition of " A\cup B" and then do it in two cases:
    1) If x\in B we are done.

    2) if x\in A then because A\subseteq B, x\in B.

    So that in either case, if x\in A\cup B then x\in B.

    All that remains is to show that B\subseteq A\cup B. To do that, if x\in B then x\in A\cup B by definition of A\cup B.
    Last edited by HallsofIvy; October 6th 2013 at 06:02 PM.
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