GATE CS 2015 Set 3 — Question 6

Let P be the statement "Alexander had conquered Persia."
Let I be the statement "Alexander turned his attention towards India."The given sentence is "Alexander turned his attention towards India, since he had conquered Persia." This implies that conquering Persia was the reason or a necessary condition for Alexander to turn his attention towards India. In logical terms, this can be interpreted as: If Alexander turned his attention towards India, then he must have conquered Persia. So, I → P.Now let's analyze the options:(A) "Alexander would not have turned his attention towards India had he not conquered Persia." This statement can be written as $\neg P \rightarrow \neg I$. This is the contrapositive of I → P. Since the contrapositive of a conditional statement is logically equivalent to the original statement, if I → P is true, then $\neg P \rightarrow \neg I$ is also true. This is a valid inference based on the interpretation of "since" implying a necessary condition.(B) "Alexander was not ready to rest on his laurels, and wanted to march to India." This statement describes Alexander's motivation or state of mind, which cannot be directly inferred from the given sentence. The sentence only states a causal link between two events.(C) "Alexander was completely in control of his army and could command it to move towards India." This statement makes an assumption about Alexander's control over his army, which is not mentioned or implied in the original sentence.(D) "Since Alexander's kingdom extended to Indian borders after the conquest of Persia, he was keen to move further." This statement introduces new information (kingdom extended to Indian borders) and an interpretation of his keenness, neither of which is directly inferable from the original sentence.Therefore, option (A) is the only logically valid inference.The final answer is $\boxed{A}$