Math Problem Statement
(p ^ q)=> p
Solution
The given logical expression is . 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:
p | q | p ∧ q | (p ∧ q) ⇒ p |
---|---|---|---|
T | T | T | T |
T | F | F | T |
F | T | F | T |
F | F | F | T |
Explanation:
- When both p and q are true (T), the expression evaluates to true (T), because is also true.
- When p is true and q is false (F), the conjunction is false, and since the implication holds true whenever the premise is false, the expression evaluates to true.
- When p is false and q is true, the conjunction is false, so the implication evaluates to true.
- 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:
- What is a tautology in logic?
- How do truth tables work in propositional logic?
- Can implications be true when the premise is false?
- What is the difference between conjunction and implication?
- How would the truth table change for the expression ?
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