2. Let A(x) be the predicate ``x Î A'', B(x) be the predicate ``x Î
B'', and let U be the universal set. Then (A Ç B)c = Ac È Bc is equivalent to
(" x Î U)
(~(A(x) Ù B(x)) « (~A(x) Ú ~ B(x)))
To prove this is true we can use a truth table:
| a | b | ~ (a Lb) | ~a V ~b | ~ (a Lb) « (~a V ~b) | |
Fill in the first two columns with the four possible combinations of truth values for a
and b. Then fill in the remaining columns, refering back to the logical
connectives from Chapter 1 if you need to.