Question

Which of the following is NOT true?

• A. $p\to (q\wedge r)\equiv (p\to q) \wedge (p \to r)$
• B. $\sim (q \leftrightarrow r)\equiv (p \wedge \sim q)\vee(\sim p \wedge q)$
• C. $(p \wedge\sim q)\leftrightarrow (p \to q)$is a tautology.
• D. $(p \to q) \wedge (q \rightarrow r) \rightarrow (p \rightarrow r)$is a tautology.

Answer 1

Answer: (C)

$(p \wedge\sim q)\leftrightarrow (p \to q)$is a tautology.

Answer 2

p
q
r
$\sim p$
$\sim q$
$\sim r$
$(p \wedge \sim q)(x)$
$(p \rightarrow q)(y)$
$(x \leftrightarrow y)$
$(q \rightarrow r)(s)$
$(p \rightarrow r)(w)$

T
T
T
F
F
F
F
T
F
T
T

T
T
F
F
F
T
F
T
F
F
F

T
F
T
F
T
F
T
F
F
T
T

T
F
F
F
T
T
T
F
F
T
T

F
T
T
T
F
F
F
T
F
T
T

F
T
F
T
F
T
F
T
F
F
T

F
F
T
T
T
F
F
T
F
T
T

F
F
F
T
T
T
F
T
F
T
T

$(y \wedge z)A$
$(A \rightarrow w)$
$(p \rightarrow z)B$
$(y \wedge w)C$
$\sim(p \leftrightarrow q)F$
$(p \rightarrow G)$
$(p\wedge q)E$
$(x \vee E)D$
$(p\wedge r)G$

T
T
T
T
F
T
F
F
T

F
T
F
F
F
F
F
F
F

F
T
T
F
T
F
F
T
T

F
T
T
F
T
F
F
T
F

T
T
T
T
T
T
T
T
T

F
T
T
T
T
T
T
T
F

T
T
T
T
F
T
F
F
F

T
T
T
T
F
T
F
F
F

(a) $(p \kappa q) \longleftrightarrow(p \rightarrow q)$
$x \longleftrightarrow y$ is a contradiction from table.
(b) $\\{(p \rightarrow q) \wedge(q \rightarrow r)\\} \rightarrow(p \rightarrow r)\\{y \wedge z\\} \rightarrow w$
$A \rightarrow w$ is a tautology from table.
(c) $p \rightarrow (q \wedge r) \equiv(p \rightarrow q) \wedge(p \rightarrow r)$
$p \rightarrow G \equiv y \wedge w$
$(p \rightarrow G) \equiv C$, represent logical equivalence from table.
(d) $\sim(p \longleftrightarrow q) \equiv(p \wedge q) \vee(-p \wedge q)$
$F \equiv x \vee E$
$F \equiv D$, represent logical equivalence from table.