Just wondering if someone could help me with this proof from a practice exam I'm studying for.
Let A and B be sets. Prove or disprove:
(B-A) union (A-B) = (A union B) - (A intersection B)
Thanks for any help.
There are probably some set rules to break this down, but look at it this way: there are only 4 possibilities that you need to check.
1) A and B are disjoint.
2) A and B intersect, but do not contain one or the other.
3) A is a subset of B.
4) B is a subset of A.
(I suppose you ought to include the possibilities were either A or B or both are the empty set, but these should be easy.)
-Dan
Hello, OntarioStud!
I will use A' for "complement of A" and 0 for the empty set.
Let A and B be sets.
Prove or disprove: .(B - A) U (A - B) = (A U B) - (A ∩ B)
Defintion of set subtraction: .A - B .= .A ∩ B'
Axiom #1: .A ∩ A' = 0
Axiom #2: .A U 0 = A
Distributive Laws: .A U (B ∩ C) = (A U B) ∩ (A U C)
. . . . . . . . . . . . . .A ∩ (B U C) = (A ∩ B) U (A ∩ C)
The right side is: .(A U B) - (A ∩ B)
. . . . . . . . . . .= .(A U B) ∩ (A ∩ B)' . Def. of Subtraction
. . . . . . . . . . .= .(A U B) ∩ (A' U B') . DeMorgan's Law
. . . . . . . . . . .= .[(A U B) ∩ A'] U [(A U B) ∩ B'] . Distr.Law
. . . . . . . . . . .= .[(A ∩ A') U (B ∩ A')] U [(A ∩ B') U (B ∩ B')] . Distr.Law
. . . . . . . . . . .= .[0 U (B ∩ A')] U [(A ∩ B') U 0] . Axiom #1
. . . . . . . . . . .= .(B ∩ A') U (A ∩ B') . Axiom #2
. . . . . . . . . . .= .(B - A) U (A - B) . Def. of Subtraction