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This statement: [(p → q)   ↔   (¬ p ∨ q) ] is tautologyGroup startsTrue or False

Question

This statement: [(p → q) ↔ (¬ p ∨ q)] is tautologyGroup starts

True or False

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Solution

To determine whether the statement (pq)(¬pq)(p \to q) \leftrightarrow (\neg p \lor q) is a tautology, we will analyze both sides using truth tables.

1. Break Down the Problem

We need to evaluate the expressions pqp \to q and ¬pq\neg p \lor q under all possible truth values for pp and qq to check if they are equivalent for all cases.

2. Relevant Concepts

  • The implication pqp \to q is defined as ¬pq\neg p \lor q.
  • A statement is a tautology if it is true for all possible truth values of the variables involved.

3. Analysis and Detail

We will construct a truth table with all possible combinations of truth values for pp and qq:

pp qq pqp \to q ¬p\neg p ¬pq\neg p \lor q (pq)(¬pq)(p \to q) \leftrightarrow (\neg p \lor q)
T T T F T T
T F F F F T
F T T T T T
F F T T T T

4. Verify and Summarize

Now, we analyze the last column:

  • In all scenarios, the expression (pq)(¬pq)(p \to q) \leftrightarrow (\neg p \lor q) evaluates to True.

Final Answer

The statement (pq)(¬pq)(p \to q) \leftrightarrow (\neg p \lor q) is a tautology. Therefore, the answer is True.

This problem has been solved

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