Q. Let P, Q, R, S and T be five statements such that:
I. If P is true, then both Q and S are true.
II. If R and S are true, then T is false.
Which of the following can be concluded?
1. If T is true, then at least one of P and R must be false.
2. If Q is true, then P is true.
Select the correct answer using the code given below:
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Correct Answer: (a) 1 only
Question from UPSC Prelims 2023 CSAT
Explanation :
Let’s analyze each statement:
1. If T is true, then at least one of P and R must be false.
- The contrapositive of 1 is: “If both P and R are true, then T is false”. This statement can be derived from the given statements as follows:
- If P is true, then Q and S are true (by I).
- If R and S are true, then T is false (by II).
- Therefore, if P and R are true, then Q, S and T are false (by transitivity).
- Since the contrapositive of 1 is true, 1 is also true.
2. If Q is true, then P is true.
According to statement I, if P is true, then both Q and S are true. However, this does not imply that if Q is true, then P must be true. It is possible for Q to be true while P is false. Therefore, this statement is false.
So, the correct answer is (a) 1 only.
