# Set operations - is this a valid proof?

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Posted by Joel on October 01, 2002 at 19:11:12:

using the following symbols:
u = union
e = "is member of"
v = intersection
| = such that
c= = "is subset of"

is this a valid proof that
(A u B) c= (A u B u C) ("A union B is a subset of A union B union C")?

given an X | X e (A u B)
then (X e A) v (X e B)
then (X e A) v (X e B) v (X e C) [this is the step that worries me, since no information is given regarding membership or non-membership in C, but there has to be some way to introduce C into the picture]

then X e (A u B u C)
therefore any (X | X e (A u B)) e (A u B u B)
so by the definition of subset, (A u B) c= (A u B u C)

The whole question seems so obvious and trivial, and this argument seems so simple that it is hard to believe that this is a legitimate proof. Also, I'm bothered by the fact that "C" seems to materialize out of thin air, but there has to be some "legal" way to get C into the argument. Is this it?

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