Expert: Sherman D. - 11/1/2005

Question
I am working on proofs. I need to prove that when A and B are sets, A union B= (A-B)union (B-A) union (A intersect B).

Answer
A U B = (A - B) U (B - A) U (A intersect B)

This shows you about what to do
http://mathforum.org/library/drmath/view/61902.html

Here is the proof using values for A and B.

A = 1,2,3,4
B = 1,3,5,6

A Intersect B = 1,3
A - B = ((1,2,3,4) - (1,3,5,6)) = 2,4
B - A = ((1,3,5,6) - (1,2,3,4)) = 5,6
(A - B) U (B - A) = (2,4) U (5,6) = 2,4,5,6

A U B = (A - B) U (B - A) U (A intersect B)
A U B = ((A - B) U (B - A)) U (A intersect B)
(1,2,3,4) U (1,3,5,6) = (2,4,5,6) U (1,3)
(1,2,3,4,5,6) = (1,2,3,4,5,6)

I just went to the site that i gave you, but instead of using names, i used numbers. Here is what they gave and here is what i put

Actors (A)
Julia Roberts = 1
Tom Hanks = 2
Helena Bonham Carter = 3
Brad Pitt = 4

Women (B)
Julia Roberts = 1
Helena Bonham Carter = 3
Sally Ride = 5
Valerie Shute = 6

In lamins terms, what this is doing is taking

(A - (A intersect B)) U (B - (A intersect B)) U (A intersect B)

If you notice, even though you took out (A intersect B) to get (A - B) and (B - A), you put the (A intersect B) right back into the problem, which gives you the A U B.

I hope this is what you were wanting.