मान लीजिए P, Q, R, S और T पाँच कथन हैं, इस प्रकार कि :
I. यदि P सत्य है, तो Q और S दोनों सत्य हैं ।
II. यदि R और S सत्य हैं, तो T असत्य है ।
निम्नलिखित में से कौन-सा/से निष्कर्ष निकाला जा सकता है/निकाले जा सकते हैं ?
1. यदि T सत्य है, तो P और R में से कम-से-कम एक अवश्य असत्य है ।
2. यदि Q सत्य है, तो P सत्य है ।
नीचे दिए गए कूट का प्रयोग कर सही उत्तर चुनिए :
सही उत्तर (A) 1 only है।
व्याख्या:
Let us analyze the given logical implications: I. P implies (Q and S), denoted as P => (Q AND S). II. (R and S) implies not T, denoted as (R AND S) => NOT T. Let us test the conclusions: 1. If T is true, then at least one of P and R must be false: Assume T is true (T = True). From II, if T is true, then the premise (R AND S) must be false (by contrapositive). Thus, NOT (R AND S) is true, which means either R is false or S is false (or both). Now, let us check P. From I, P => S. If P were true, then S must be true. But does this force P and R to be false? If P is true, S is true. If S is true, and R is false, then (R AND S) is false, which satisfies II. Here P is true and R is false. So P is not necessarily false! Thus, conclusion 1 does not necessarily hold. 2. If Q is true, then P is true: Let us check the contrapositive of I. I states P => (Q AND S). The contrapositive is NOT (Q AND S) => NOT P. This does not directly prove Q => P. However, let us use formal logic: Can Q be true while P is false? Yes, implication P => Q does not mean Q => P. Wait, let us re-verify: Does Q true imply P true? No, implication is one-way. Let us check if both conclusions are false, making the answer 'Neither 1 nor 2'. In propositional logic, P => Q & S means if P is true, Q is true. But Q being true gives no information about P. Hence neither 1 nor 2 can be logically deduced.
In English (Question & Model Answer)
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 :
The correct answer is (A) 1 only.
Explanation:
Let us analyze the given logical implications: I. P implies (Q and S), denoted as P => (Q AND S). II. (R and S) implies not T, denoted as (R AND S) => NOT T. Let us test the conclusions: 1. If T is true, then at least one of P and R must be false: Assume T is true (T = True). From II, if T is true, then the premise (R AND S) must be false (by contrapositive). Thus, NOT (R AND S) is true, which means either R is false or S is false (or both). Now, let us check P. From I, P => S. If P were true, then S must be true. But does this force P and R to be false? If P is true, S is true. If S is true, and R is false, then (R AND S) is false, which satisfies II. Here P is true and R is false. So P is not necessarily false! Thus, conclusion 1 does not necessarily hold. 2. If Q is true, then P is true: Let us check the contrapositive of I. I states P => (Q AND S). The contrapositive is NOT (Q AND S) => NOT P. This does not directly prove Q => P. However, let us use formal logic: Can Q be true while P is false? Yes, implication P => Q does not mean Q => P. Wait, let us re-verify: Does Q true imply P true? No, implication is one-way. Let us check if both conclusions are false, making the answer 'Neither 1 nor 2'. In propositional logic, P => Q & S means if P is true, Q is true. But Q being true gives no information about P. Hence neither 1 nor 2 can be logically deduced.