Which of the following statements is a tautology?
Show Hint
To check if a statement is a tautology, simplify the logical expression step-by-step using equivalence rules (e.g., distributive, associative, and De Morgan’s laws) and test it for all possible truth values of the variables.
\(p \vee( p \wedge q )\)
\(( p \wedge( p \rightarrow q )) \rightarrow \sim q\)
\(( p \wedge q ) \rightarrow(\sim( p ) \rightarrow q )\)
\(p \rightarrow( p \wedge( p \rightarrow q ))\)
The Correct Option is C
Solution and Explanation
Step 1: Analyze Each Statement
(i) \( p \to (p \land (p \to q)) \):
\[p \to (p \land (p \to q)) \equiv \neg p \lor (p \land (p \to q)).\]
Using distributive property:
\[\neg p \lor (p \land (p \to q)) = (\neg p \lor p) \land (\neg p \lor (p \to q)).\]
This simplifies to:
\[\text{True} \land (\neg p \lor (\neg p \lor q)).\]
\[= \neg p \lor q.\]
This is not always true, so it is not a tautology.
(ii) \( (p \land q) \to \neg (p \to q) \):
\[(p \land q) \to \neg (p \to q) \equiv \neg (p \land q) \lor \neg (\neg p \lor q).\]
\[= (\neg p \lor \neg q) \lor (p \land \neg q).\]
This simplifies to:
\[\text{True for all cases}.\]
Hence, it is tautology
(iii) \( (p \land (p \to q)) \to \neg q \):
\[(p \land (p \to q)) \to \neg q \equiv \neg (p \land (p \to q)) \lor \neg q.\]
\[= (\neg p \lor \neg (p \to q)) \lor \neg q.\]
\[= (\neg p \lor (\neg p \land \neg q)) \lor \neg q.\]
\[= (\neg p \lor \neg q) \lor \neg q.\]
This is not always true, so it is not a tautology
(iv) \( p \lor (p \land q) \)
\[p \lor (p \land q) \equiv p.\]
This is not always true, so it is not a tautology.
Conclusion
The statement \((p \land q) \to \neg (p \to q)\) is a tautology.
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Concepts Used:
Mathematical Reasoning
Mathematical reasoning or the principle of mathematical reasoning is a part of mathematics where we decide the truth values of the given statements. These reasoning statements are common in most competitive exams like JEE and the questions are extremely easy and fun to solve.
Types of Reasoning in Maths:
Mathematically, reasoning can be of two major types such as:
- Inductive Reasoning - In this, method of mathematical reasoning, the validity of the statement is examined or checked by a certain set of rules, and then it is generalized. The principle of mathematical induction utilizes the concept of inductive reasoning.
- Deductive Reasoning - The principle is the opposite of the principle of induction. Contrary to inductive reasoning, in deductive reasoning, we apply the rules of a general case to a provided statement and make it true for particular statements. The principle of mathematical induction utilizes the concept of deductive reasoning.