Posted by T.Gracken on October 02, 2002 at 11:55:25:
In Reply to: Re: Set operations - CORRECTED - please look at this one instead posted by Joel on October 02, 2002 at 10:06:30:
: Not exactly.
yes exactly.
The definition states that the union of the sets A and B is the set that contains those elements that are either in A or in B or in both. What you are stating is probably a theorem derived from the definition. Anyway, my discrete structures professor and the author of the textbook both seem to agree that the statement (A u B) c= (A u B u C)
: needs to be proven, even though it is painfully obvious that it is true. I guess the point is to teach how to construct a formal proof, step by logical step.
let x be in AUB. then AUB is a set. Let C be a set. then Then if x is in (AUB)UC, then by (your own) definition, x is either in (AUB) or x is in C or x is in both. period. Since we know (if you like: "by construction") x is in (AUB) we are done. period. straight from the definition.
However, if your professor wants a proof of a trivial statement, let D = AUB and let x be in D. then by definition, x is in DUC for any set C. And by substitution, X is in (AUB)UC
: Anyway, I did find something that allows the step that was worrying me. Since "P -> P v Q" is a tautology, (P v Q) -> (P v Q) v X (or any other proposition), so (X e A) v (X e B) => (X e A) v (X e B) v (X e C)