Math Problem Statement

(p ^ q)=> p

Solution

The given logical expression is (pq)p(p \land q) \Rightarrow p. This can be interpreted as:

"If p and q are both true, then p must also be true."

Truth Table Analysis:

Let's create a truth table to analyze the expression:

pqp ∧ q(p ∧ q) ⇒ p
TTTT
TFFT
FTFT
FFFT

Explanation:

  1. When both p and q are true (T), the expression evaluates to true (T), because pp is also true.
  2. When p is true and q is false (F), the conjunction pqp \land q is false, and since the implication \Rightarrow holds true whenever the premise is false, the expression evaluates to true.
  3. When p is false and q is true, the conjunction pqp \land q is false, so the implication evaluates to true.
  4. When both p and q are false, the conjunction is false, and so the implication is also true.

Thus, the entire expression is always true, making it a tautology.

Do you have any questions or would you like more details?


Relative questions:

  1. What is a tautology in logic?
  2. How do truth tables work in propositional logic?
  3. Can implications be true when the premise is false?
  4. What is the difference between conjunction and implication?
  5. How would the truth table change for the expression pqp \Rightarrow q?

Tip: In logic, an implication is always true when the premise is false, regardless of the conclusion.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Implication
Conjunction
Tautology

Formulas

Truth Table for (p ∧ q) ⇒ p

Theorems

Tautology in Propositional Logic
Implication Theorem

Suitable Grade Level

Grades 9-12