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) | |
| T | T | F | F | T | |
| T | F | T | T | T | |
| F | T | T | T | T | |
| F | F | T | T | T |
Since ~ (a Lb) « (~a V ~b) is a tautology, we know that
the subset relation (A Ç B)c = Ac È Bc is true.