Math Problem Statement

answer the true one

Let p: 15 is an odd number. Let q: 21 is a prime number. Which of the following statement is correct about p ∨ q, p ∧ q, p ⇒ q, and ¬q ⇒ p?

Solution

Let’s analyze the provided question and solve it step-by-step:

Problem Summary:

  1. Let pp: "15 is an odd number"
    This is true because 15 is indeed an odd number.

  2. Let qq: "21 is a prime number"
    This is false because 21 is not a prime number (it can be factored as 3×73 \times 7).

We now evaluate each logical statement based on pp being true and qq being false.


Logical Statements to Evaluate:

  1. pqp \lor q (p OR q):

    • OR is true if at least one of pp or qq is true.
    • Here, pp is true and qq is false.
    • pq=truep \lor q = \text{true}.
  2. pqp \land q (p AND q):

    • AND is true only if both pp and qq are true.
    • Here, pp is true but qq is false.
    • pq=falsep \land q = \text{false}.
  3. p    qp \implies q (p implies q):

    • Implication p    qp \implies q is false only if pp is true and qq is false.
    • Here, pp is true and qq is false.
    • p    q=falsep \implies q = \text{false}.
  4. ¬q    p\neg q \implies p (NOT q implies p):

    • First, evaluate ¬q\neg q (NOT q). Since qq is false, ¬q\neg q is true.
    • Implication ¬q    p\neg q \implies p is true if ¬q\neg q is true or pp is true.
    • Both ¬q\neg q and pp are true.
    • ¬q    p=true\neg q \implies p = \text{true}.

Correct Statement:

From the evaluation above:

  • pq=truep \lor q = \text{true}
  • pq=falsep \land q = \text{false}
  • p    q=falsep \implies q = \text{false}
  • ¬q    p=true\neg q \implies p = \text{true}

Thus, the correct statements are pqp \lor q and ¬q    p\neg q \implies p.

Let me know if you’d like further clarification!


Related Questions:

  1. What is the difference between conjunction (\land) and disjunction (\lor) in logic?
  2. Can you explain why p    qp \implies q is false in this case?
  3. How is the truth value of an implication (    \implies) determined in logic?
  4. What are prime numbers, and why is 21 not prime?
  5. How would the answers change if qq were true instead?

Tip:

In logic, implications p    qp \implies q are tricky because they are true except when pp is true and qq is false. Always evaluate implications carefully!

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Math Problem Analysis

Mathematical Concepts

Logic
Propositional Calculus
Truth Tables

Formulas

Logical OR (p ∨ q)
Logical AND (p ∧ q)
Implication (p ⇒ q)
Negation (¬q)
Implication (¬q ⇒ p)

Theorems

Truth Table for Propositional Logic
Evaluation of Logical Connectives

Suitable Grade Level

Grades 9-12