GATE CS 2014 Set 1 — Question 63
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Engineering Mathematics → Mathematical Logic → Propositional Logic
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Question
Which one of the following propositional logic formulas is TRUE when exactly two of and are TRUE?
Correct answer
(B) ((p rightarrow q) ∧ r) ∨ (p ∧ q ∧ r)
Solution
We need a formula that evaluates to TRUE if and only if exactly two of the variables are TRUE. The possible truth assignments where exactly two are TRUE are:
Term corresponds exactly to case 1 ( true, false).
Term involves and . Since covers case 1, must cover cases 2 and 3.In cases 2 and 3, is TRUE. Also, in case 2 () and case 3 (), the values of and are different. The expression is TRUE when and are the same, and FALSE when they are different. Therefore, is TRUE when and are different.So, the term captures exactly cases 2 and 3 (where and is TRUE).Combining these with OR ():
This formula is TRUE for cases 2, 3 (first part) and case 1 (second part).This matches Option (B).
1.
2.
3.
Let's analyze the structure of the options, which generally have the form or .Term corresponds exactly to case 1 ( true, false).
Term involves and . Since covers case 1, must cover cases 2 and 3.In cases 2 and 3, is TRUE. Also, in case 2 () and case 3 (), the values of and are different. The expression is TRUE when and are the same, and FALSE when they are different. Therefore, is TRUE when and are different.So, the term captures exactly cases 2 and 3 (where and is TRUE).Combining these with OR ():
This formula is TRUE for cases 2, 3 (first part) and case 1 (second part).This matches Option (B).
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