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Math Help - Help--is this even an identity?

  1. #1
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    Help--is this even an identity?

    \text{Show } Z - (Y - X) = X \cup (Z - Y).

    I've tried this to no avail. The farthest I've gotten is Z - (Y - X) = Z \cap (Y' \cup X), and I'm not sure how in the world to get that to equal the right side. Is there a mistake somewhere in the question? It looks like there's a Z missing on the right side of the original statement. (Yes, I looked at the book about ten times to make sure that I copied it right.)
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  2. #2
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    Re: Help--is this even an identity?

    It's not true. Here is a counter-example:

    Let X = {a,b,c,d}, Y = {b,c,e,f}, Z = {c,d,f,g}

    Then Z - (Y - X) = {c,d,f,g} - {e,f} = {c,d,g} whereas:

    X U (Z - Y) = {a,b,c,d} U {d,g} = {a,b,c,d,g}, and these two sets are clearly unequal.

    It is probably a typo, perhaps the author intended:

    Z - (Y - X) is a subset of X U (Z - Y), or:

    Z - (Y - X) = (X ∩ Y ∩ Z) U (Z - Y), it's hard to say.
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  3. #3
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    Re: Help--is this even an identity?

    OK good, it wasn't just my imagination.
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