# Boolean equation Indexing exercise

By the method of indexing, prove the following Boolean equation to be valid:

(A ∪ B)′ ∩ C = (C ∩ A′) ∪ (C ∩ B′)

What I tried to do - but somehow it doesn't go right, where did I made a mistake, Please?

A - 1,2,3,4

B - 1,2,5,6

C - 1,3,5,7

1) A ∪ B = (1,2,3,4,5,6) so (A ∪ B)′ =(7,8)

(7,8) ∩ (1,3,5,7) = (7)

2) (C ∩ A′) = (1,3,5,7) ∩ (5,6,7,8) = (5,7)

(C ∩ B′) = (1,3,5,7) ∩ (3,4,7,8) = (3,7)

so (5,7) ∪ (3,7) = (3,5,7)

3) So it comes to (7) = (3,5,7) what surely can't be right?

• I believe the initial expression is incorrect. Are you sure it's not (A ∪ B)′ ∩ C = (C ∩ A′) ∩ (C ∩ B′)? (Notice that I put intersection instead of union between the parenthesis.)
– user3017
Dec 8 '16 at 20:02
• This is as it is in the book, ∪ (page 15, Smullyan, A Beginners Guide to Mathematical Logic) - but I thought the same thing that perhaps it is typo? Dec 8 '16 at 20:06
• I think it must be a typo, because your proof looks correct. I also drew the diagram which indicated the same thing.
– user3017
Dec 8 '16 at 20:08
• @Pé de Leão , Thank You, I was just afraid that perhaps I am doing something wrong :) Dec 8 '16 at 20:10
• Just to reinforce that Pe de Leao's is correct, the only thing that is changed by the negation of (A u B), is what's inside the parenthesis [A u B] NOT what is between the parenthesis and C [( ) n C]. So, the correct statement would be, (A u B)' n C = (A' n C) n (B' n C). Dec 14 '16 at 21:03

As the comments note there is an error in the problem. To check the problem, one can use truth tables. Here is a truth table for the original claim:

(A ∪ B)′ ∩ C = (C ∩ A′) ∪ (C ∩ B′) Here is a truth table for the modified claim:

(A ∪ B)′ ∩ C = (C ∩ A′) ∩ (C ∩ B′) Michael Rieppel. Truth Table Generator. Generated on May 12, 2019 at https://mrieppel.net/prog/truthtable.html