Chapter 1: Propositions and Predicates &;;l 33
(d) Ram ran home.
(e) An even number is a prime number.
(f) 10 is a root of the equation .~ - lO02x + 10000 = 0
(g) Go home and take rest.
1.2 Express the following sentence in symbolic form: For any two numbers
a and h, only one of the following holds: a < b, a = h, and a > h.
1.3 The truth table of a connective called Exclusive OR (denoted by v) is
shown in Table 1.18.
TABLE 1.18 Truth Table for Exclusive OR
p
Q
P v Q
T
T
F
T
f=
T
F
T
T
F
F
F
Give an example of a sentence in English (i) in which Exclusive OR
is used, (ii) in which OR is used. Show that v is associative,
commutative and distributive over I\.
1.4 Find two connectives, using which any other connective can be
desClibed.
1.5 The connective Ni\ND denoted by i (also called the Sheffer stroke) is
defined as follO\l/s: P i Q = . . . . . , (P ;\ Q). Show that every connective
can be expressed in terms of NAND.
1.6 The connective NOR denoted by 1 (also called the Peirce arrow) is
defined as follows: P 1 Q = ....., (P v Q). Show that every connective
can be expressed in terms of NOR.
1.7 Construct the truth table for the following:
(a) (P v Q) => ((P v R) => (R v Q)
(b) (p v (Q => R) <=::? ((P v .....,R) => Q)
1.8 Prove the follmving equivalences:
(a) (....., P => (-, P => (-, P ;\ Q») == P v Q
(b) P == (P v Q) ;\ (P v ....., Q)
(c) ....., (P <=::? Q) == (P 1\ ....., Q) V (-, P /\ Q)
1.9 Prove the logical identities given in Table 1.11 using truth tables.
1.10 Show that P => (Q => (R => (....., P => (-, Q => ....., R») is a tautology.
1.11 Is (P => ....., Pl => ....., P (i) a tautology. (ii) a contradiction. (iii) neither
a tautology nor a contradiction?
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