GATE CS 2017 Set 1 — Question 29
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Engineering Mathematics → Mathematical Logic → Propositional Logic
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Question
Let and be propositions and the expression be a contradiction. Then, the expression is
Correct answer
(D) always TRUE when q is TRUE.
Solution
We are given that is a contradiction, which means it evaluates to FALSE.
For an implication
Here, and .
So, we have:
Substitute :
Since is always TRUE regardless of the value of , the expression simplifies to:
Since is logically equivalent to , the value of the expression depends entirely on .Checking the options:
(A) It is not a tautology because can be FALSE.
(B) It is not a contradiction because can be TRUE.
(C) If is FALSE, is TRUE (consistent with premise), but could still be FALSE. So the expression is not necessarily TRUE.
(D) If is TRUE, then the expression (which is equivalent to ) is TRUE. This holds.Thus, the expression is always TRUE when is TRUE.
For an implication
A → B to be FALSE, must be TRUE and must be FALSE.Here, and .
So, we have:
1.
2.
We need to evaluate the expression .Substitute :
Since is always TRUE regardless of the value of , the expression simplifies to:
Since is logically equivalent to , the value of the expression depends entirely on .Checking the options:
(A) It is not a tautology because can be FALSE.
(B) It is not a contradiction because can be TRUE.
(C) If is FALSE, is TRUE (consistent with premise), but could still be FALSE. So the expression is not necessarily TRUE.
(D) If is TRUE, then the expression (which is equivalent to ) is TRUE. This holds.Thus, the expression is always TRUE when is TRUE.
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