# Help--is this even an identity?

• Dec 30th 2013, 10:42 PM
phys251
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.)
• Dec 30th 2013, 11:32 PM
Deveno
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.
• Dec 31st 2013, 03:13 PM
phys251
Re: Help--is this even an identity?
OK good, it wasn't just my imagination.