Expert: Sherry Wallin - 4/18/2010

Question
QUESTION: A, B, and C are subsets of a set S. Prove the following
set identity using the basic set of identities (p. 197) Justify each
step with the set identity you used.
(A ∪ C) ∩ (A ∩ B) ∪ (C ∩ B) = A ∩ B

ANSWER: Ashley~
    I need a couple of things to answer this question. One I need the 'basic set of identities' that you were given and two I need to know what (C ∩ B) represents.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: I made a mistake this should be it

(A ∪ C) ∩ [(A ∩ B) ∪ (C' ∩ B)] = A ∩ B

Answer
Hi Ashley~
    I still don't know what your basic set of identities is but this is how I would do the problem. Incidentally no matter how I do it, the result is NOT A ∩ B, it is just A:

(A ∪ C) ∩ [(A ∩ B) ∪ (C' ∩ B)]
= [(A ∪ C) ∩ (A ∩ B)] ∪ [(A ∪ C) ∩ (C' ∩ B)] distribute the intersection over the union
= A ∪ [(A ∪ C) ∩ (C' ∩ B)] the intersection of (A ∪ C) ∩ (A ∩ B) is just A
= [A ∪ (A ∪ C)] ∩ [A ∪ (C' ∩ B)] distribute the union over the intersection
= (A ∪ C) ∩ [A ∪ (C' ∩ B)]  the union of A with thee union of A and C is just the union of A and C
= (A ∪ C) ∩ [(A ∪ C') ∩ (A ∪ B)] distribute the union over the intersection
= (A ∪ C) ∩ A the intersection of A and anything with A and anything is just A
= A ditto from above

Math Prof

PS You can also prove the statement (A ∪ C) ∩ [(A ∩ B) ∪ (C' ∩ B)] = A by choosing an element x in (A ∪ C) ∩ [(A ∩ B) ∪ (C' ∩ B)] and showing it is just in A and then choose the element x in A and show it is also in (A ∪ C) ∩ [(A ∩ B) ∪ (C' ∩ B)] but that doesn't seem to be what your teacher wants you to do at this point.