A.l The Logic of Propositions 587ALTERNATIVE WAYS OF SAYING P::::} QThe English language provides a variety of expressions, all of which are
equivalent to P::::} Q:
If P, then Q.
If P , Q.
P implies Q.
Q if P.
P only if Q.
P is a sufficient condition for Q.
P is sufficient for Q.
Q is a necessary condition for P.
Q is necessary for P.Examples A.1.8 The following are statements of the form P::::} Q. They are
all equivalent (assuming xis a known number).
(a) If x > 5 then x > l.
(b) If x > 5, x > l.
(c) x > 5 implies x > l.
( d) x > 1 if x > 5.
( e) x > 5 only if x > l.
(f) x > 5 is a sufficient condition for x > l.
(g) x > 5 is sufficient for x > 1.
(h) x > 1 is a necessary condition for x > 5.
(i) x > 1 is necessary for x > 5. DNotes: (a) x = 1 is a sufficient, but not necessary, condition, for x^2 = l.
(b) cos x = 1 is a necessary, but not sufficient, condition for x = 0.
( c) In any triangle, having two congruent sides is a necessary and sufficient
condition for having two congruent angles.
Definition A.1.9 The bi-implication "P if and only if Q" is symbolized
P {::} Q and is defined by the truth-table:
Table A.4
p Q p {::} Q
T T T
T F F
F T F
F F T