Determine the truth value of the statement (p V q) → q, given that p is true and q is false.

Understand the Problem

The question is asking to determine the truth value of the logical statement (p ∨ q) → q, given specific truth values for p and q. This involves understanding and applying logical operations.

Answer

False

Steps to Solve

Identify the given truth values We are given that ( p ) is true and ( q ) is false:

( p = \text{True} )
( q = \text{False} )

Evaluate ( p \lor q ) The logical disjunction (OR) is true if at least one of the operands is true. $$ p \lor q = \text{True} \lor \text{False} = \text{True} $$
Evaluate ( (p \lor q) \rightarrow q ) Now we evaluate the implication (→). An implication is false only if the first operand is true and the second operand is false: $$ (p \lor q) \rightarrow q = \text{True} \rightarrow \text{False} $$
Determine the truth value of the implication Since the first part is true and the second part is false, the implication evaluates to false: $$ \text{True} \rightarrow \text{False} = \text{False} $$

The truth value of the statement ( (p \lor q) \rightarrow q ) is ( \text{False} ).

More Information

The logical implication states that if ( p \lor q ) is true, then ( q ) must also be true. Here, ( p \lor q ) is true, but ( q ) is false, making the whole statement false. This is a fundamental concept in propositional logic.

Tips

One common mistake is to misinterpret the truth value of implications. Remember that an implication is false only when the first part is true and the second is false.

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