Math Problem Statement

If p and q are false, what is the truth value of ¬ (p Λ q)?

Solution

To determine the truth value of the expression ¬(pq)\neg (p \land q), we need to follow these steps:

  1. Understand the components:

    • pp and qq are both false (FF).
    • \land is the logical "and" operator, which results in true only if both operands are true.
    • ¬\neg is the logical "not" operator, which inverts the truth value.
  2. Evaluate the expression inside the parentheses:
    pqp \land q
    Since both pp and qq are false, pqp \land q is also false.

    pq=FF=Fp \land q = F \land F = F

  3. Apply the negation: ¬(pq)=¬F=T\neg (p \land q) = \neg F = T

Thus, the truth value of ¬(pq)\neg (p \land q) is true.

Would you like more details, or do you have any questions?

Here are 5 related questions to expand your understanding:

  1. What is the truth value of pqp \lor q when both pp and qq are false?
  2. How does the truth table of ¬(pq)\neg (p \lor q) compare with ¬(p)¬(q)\neg (p) \land \neg (q)?
  3. What is the truth value of ¬(pq)\neg (p \lor q) if pp is true and qq is false?
  4. Can you explain De Morgan's laws using the given expression?
  5. What happens to the truth value of ¬(pq)\neg (p \land q) if pp is true and qq is false?

Tip: When working with logical expressions, it's helpful to construct truth tables to visualize how different operators affect the truth values.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Logical Operators

Formulas

¬(p Λ q) = ¬F = T

Theorems

De Morgan's Laws

Suitable Grade Level

Grades 9-12