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
Answer for screen readers
The truth value of the statement ( (p \lor q) \rightarrow q ) is ( \text{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} )
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Evaluate ( p \lor q ) The logical disjunction (OR) is true if at least one of the operands is true. $$ p \lor q = \text{True} \lor \text{False} = \text{True} $$
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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: $$ (p \lor q) \rightarrow q = \text{True} \rightarrow \text{False} $$
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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.