4,668 views

Choose the correct choice(s) regarding the following proportional logic assertion $S$:

$$S: (( P \wedge Q) \rightarrow R) \rightarrow (( P \wedge Q) \rightarrow (Q \rightarrow R))$$

1. $S$ is neither a tautology nor a contradiction
2. $S$ is a tautology
3. $S$ is a contradiction
4. The antecedent of $S$ is logically equivalent to the consequent of $S$

Antecedent of $S: (P \wedge Q) \to R$

$\qquad \equiv \neg ( P\wedge Q) \vee R$

$\qquad \equiv \neg P \vee \neg Q \vee R$

Consequent of $S : (P \wedge Q) \to (Q \to R)$

$\qquad\equiv (P \wedge Q) \to (\neg Q \vee R)$

$\qquad\equiv \neg( P\wedge Q) \vee (\neg Q \vee R)$

$\qquad\equiv \neg P \vee \neg Q \vee (\neg Q \vee R)$

$\qquad\equiv \neg P \vee \neg Q \vee R$

Antecedent of $S$ is equivalent to Consequent of $S.$ Hence Option D is right.

$A \to A$ is a Tautology. Hence options A and C are wrong and option B is right.

and A→¬A, is contingency, right?
yes

Options $(b)$ and $(d)$.

Refer to the image below.

Shouldn’t it be a tautology?

The left hand side and right hand side are saying the same thing.
Correct. Edited.
Very interesting approach. Thanks for sharing!!

A tautology is a proposition that is always true for every value of its propositional variables.

A contradiction is a proposition that is always false for every value of its propositional variables.

Logically equivalent: Compound propositions with the same truth value in all possible cases are logically equivalent.

in another way, the compound propositions $p$ and $q$ are called logically equivalent if $p\leftrightarrow q$ is a tautology.

 $P$ $Q$ $R$ $P\land Q$ $((P\land Q)\rightarrow R)$ $(Q\rightarrow R)$ $((P\land Q)\rightarrow(Q\rightarrow R))$ $((P\land Q)\rightarrow R)\rightarrow((P\land Q)\rightarrow(Q\rightarrow R))$ $T$ $T$ $T$ $T$ $T$ $T$ $T$ $T$ $T$ $T$ $F$ $T$ $F$ $F$ $F$ $T$ $T$ $F$ $T$ $F$ $T$ $T$ $T$ $T$ $T$ $F$ $F$ $F$ $T$ $T$ $T$ $T$ $F$ $T$ $T$ $F$ $T$ $T$ $T$ $T$ $F$ $T$ $F$ $F$ $T$ $F$ $T$ $T$ $F$ $F$ $T$ $F$ $T$ $T$ $T$ $T$ $F$ $F$ $F$ $F$ $T$ $T$ $T$ $T$

It’s clearly visible from the above truth table that $S$ is a tautology as all its values are true. so option $B$ is true and $A$ is false.

$S$ is not a contradiction because all its values are true. so option $C$ is false.

The antecedent of $S$ is logically equivalent to the consequent of $S$, this option is true.

From the above truth table, we can see that $((P\land Q)\rightarrow R)$ $\equiv$ $((P\land Q)\rightarrow(Q\rightarrow R))$ and their truth tables are the same.

Option $B,D$ are correct.

### 1 comment

…..….….….….…..…..

by