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.