Chapter 1: Propositions and Predicates !ml 27
TABLE 1.16 Truth Table of Example 1.26
P
Q
R
Q<=;R
-,P
-,Q
-,P<=;-,Q
ex
T
T
T
T
F
F
T
T
T
T
F
F
F
F
T
F
T
F
T
F
F
T
F
T
T
F
F
T
F
T
F
F
F
T
T
T
T
F
F
F
F
T
F
F
T
F
F
T
F
F
T
F
T
T
T
F
F
F
F
T
T
T
T
T
EXAM PLE 1.27
Prove that: ex = «P :::::} (Q v R)) !\ (---, Q)) :::::} (P :::::} R) is a tautology.
Solution
Let f3 = (P :::::} (Q v R)) !\ (---, Q)
The truth table is constructed as shown m Table 1.17. From the truth
table, we conclude that ex is a tautology.
TABLE 1.17 Truth Table of Example 1.27
P
Q
R
-,Q
QvR
P => (Q v R)
f3
P=>R
ex
T
T
T
F
T
T
F
T
T
T
T
F
F
T
T
F
F
T
T
F
T
T
T
T
T
T
T
T
F
F
T
F
F
F
F
T
F
T
T
F
T
T
F
T
T
F
T
F
F
T
T
F
T
T
F
F
T
T
T
T
T
T
T
F
F
F
T
F
T
T
T
T
EXAMPLE 1.28
State the converse, opposite and contrapositive to the following statements:
(a) If a triangle is isoceles, then two of its sides are equal.
(b) If there is no unemployment in India, then the Indians won't go to
the USA for employment.
Solution
If P :::::} Q is a statement, then its converse, opposite and contrapositive
st~tements are, Q :::::} P, ---, P :::::} ---, Q and ---, Q :::::} ---, P, respectively.
(a) Converse-If two of the sides of a triangle are equal, then the triangle
is isoceles.
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