Tautology Concept- GATE 2025- Previous year paper discrete mathematics consider the following

Ali Seron Jason

2 months ago

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A tautology in propositional logic is a formula that is true in every possible interpretation, meaning it always evaluates to true regardless of the truth values of its constituent propositions. For example, the statement $\vee \neg P$ (where $∨\vee$ denotes logical OR and $¬\neg$ denotes logical NOT) is a tautology because either $P$ is true or $¬P\neg P$ is true, covering all possible scenarios.

In the context of the GATE 2025 examination, questions related to tautologies often require determining whether a given propositional formula is always true. For instance, consider the following question from the GATE CSE 2025 Set 1 paper:

Which one of the following well-formed formulae is a tautology?

$\rightarrow Q) \rightarrow (\neg Q \rightarrow \neg P)$
$\rightarrow Q) \rightarrow (Q \rightarrow P)$
$\rightarrow (Q \rightarrow P)$
$\wedge Q) \rightarrow (P \vee Q)$

Solution:

Option 1: $\rightarrow Q) \rightarrow (\neg Q \rightarrow \neg P)$
- This is the contrapositive law, which states that a conditional statement is logically equivalent to its contrapositive. Therefore, this formula is a tautology.
Option 2: $\rightarrow Q) \rightarrow (Q \rightarrow P)$
- This suggests that if $P$ implies $Q$ , then $Q$ implies $P$ . This is not necessarily true in all cases, so it is not a tautology.
Option 3: $\rightarrow (Q \rightarrow P)$
- This indicates that if $P$ is true, then $Q$ implies $P$ . Regardless of $Q$ ‘s value, $P$ remains true if $P$ is true. Thus, this is a tautology.
Option 4: $\wedge Q) \rightarrow (P \vee Q)$
- If both $P$ and $Q$ are true, then at least one of them is true. This is always the case, making this formula a tautology.

Therefore, options 1, 3, and 4 are tautologies. However, since the question asks for the one well-formed formula that is a tautology, and multiple options satisfy this condition, it might be a misinterpretation or misprint in the question. Typically, such questions are designed to have a single correct answer. Candidates should carefully analyze each option and refer to authoritative sources or errata provided by the examination authorities for clarification.

For more practice on such topics, you can refer to previous years’ GATE questions on mathematical logic and tautologies. Websites like ExamSIDE provide a collection of past GATE questions with solutions, which can be beneficial for understanding the pattern and difficulty level of the questions.