1. Let P(x) be the predicate ``x Î
P'', Q(x) be the predicate ``x Î Q'', and let U be
the universal set. Then P Í P È Q is equivalent to
(" x Î U)
(P(x) ® (P(x) Ú Q(x))).
To prove this is true we can use a truth table:
| p | q | p Ú q | p ® (p Ú q) | |
| T | T | T | T | |
| T | F | T | T | |
| F | T | T | T | |
| F | F | F | T |
Since p ® (p Ú q) is a tautology, we know that
the subset relation P Í P È Q is true.