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) | |
Fill in the first two columns with the four possible combinations of truth values for p
and q. Then fill in the remaining columns, refering back to the logical
connectives from Chapter 1 if you need to.