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  1. #1
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    proof

    if A union B = A union C, then B=C.

    proof: Assume that A union B = A union C. Suppose x is in B. Then x is in A U B. By the assumption x is also in A U C. But we did not assume x was in A so x must be in C. Thus,we have shown B is a subset of C and by a parallel argument C is a subset of B

    is this right
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by leinadwerdna View Post
    if A union B = A union C, then B=C.

    proof: Assume that A union B = A union C. Suppose x is in B. Then x is in A U B. By the assumption x is also in A U C. But we did not assume x was in A so x must be in C. Thus,we have shown B is a subset of C and by a parallel argument C is a subset of B

    is this right
    Not quite. Let A=\{1, 2, 3\}, B=\{3,4\}. Can you perhaps see the problem in your proof by looking at these examples?

    However, you can quite easily tweak your proof to make it valid, taking into account the problem I've hinted at.
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  3. #3
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    Quote Originally Posted by leinadwerdna View Post
    if A union B = A union C, then B=C
    Quote Originally Posted by Swlabr View Post
    However, you can quite easily tweak your proof to make it valid, taking into account the problem I've hinted at.
    No! No matter how one tweaks it, we can’t prove something that is false to be true.
    \begin{gathered}<br />
  A = \left\{ {1,2,3} \right\},\,B = \left\{ {1,4} \right\}\;\& \,C = \left\{ {3,4} \right\} \hfill \\<br />
  \left\{ {1,2,3,4} \right\} = A \cup B = A \cup C\text{ but }B \ne C \hfill \\ <br />
\end{gathered}
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  4. #4
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    thank you
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