Consider : Statement - I : [( p q ) ( p q )] is a fallacy.

Question

Consider :

Statement - I : $$\left( {p \wedge  \sim q} \right) \wedge \left( { \sim p \wedge q} \right)$$ is a fallacy.

Statement - II :$$\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to  \sim p} \right)$$ is a tautology.

Accepted Answer

Statement - I : $$(p \wedge  \sim q) \wedge ( \sim p \wedge q)$$

p
    q
    $$ \sim p$$
    $$ \sim q$$
    $$p \wedge  \sim q$$

$$ \sim p \wedge q$$

$$(p \wedge  \sim q) \wedge ( \sim p \wedge q)$$

T
    T
    F
    F
    F
    F
    F

T
    F
    F
    T
    T
    F
    F

F
    T
    T
    F
    F
    T
    F

F
    F
    T
    T
    F
    F
    F

Hence, statement-I is a fallacy.

Statement - II : $$(p \to q) \leftrightarrow ( \sim q \to  \sim p)$$

p
    q
    $$ \sim p$$
    $$ \sim q$$
    $$p \to q$$

$$ \sim q \to  \sim p$$

$$(p \to q) \leftrightarrow ( \sim q \to  \sim p)$$

T
    T
    F
    F
    T
    T
    T

T
    F
    F
    T
    F
    F
    T

F
    T
    T
    F
    T
    T
    T

F
    F
    T
    T
    T
    T
    T

Hence, statement - II is a tautology.

Now, statement-II does not depend on statement-I.

Solutions · 1