GATE CS 2017 Set 1 Q29 — Linear Algebra: Matrices, Determinants, System of Linear Equations

An implication $A \rightarrow B$ is false if and only if $A$ is True and $B$ is False.

Given that the expression $(p \rightarrow q) \rightarrow r$ is a contradiction (always False), we must have:

1. $r$ is False.
2. $p \rightarrow q$ is True.

Now, let us evaluate the target expression $(r \rightarrow p) \rightarrow q$ using these conditions:

• Since $r$ is False, the antecedent $r \rightarrow p$ simplifies to $\text{False} \rightarrow p$, which is always True.
• Thus, the entire expression $(r \rightarrow p) \rightarrow q$ simplifies to $\text{True} \rightarrow q$, which is logically equivalent to $q$.

Since the expression simplifies directly to $q$, it is always True when $q$ is True.