Math Problem Statement
Help me the answer
Construct a truth table for the statement: (~p ∧ q) → (p ∧ ~q). Then determine if the statement is a tautology.
Solution
To solve this problem, we need to construct a truth table for the given logical expression:
Where:
- means "not " (negation of ),
- is the logical AND,
- is the logical implication.
Steps to Construct the Truth Table:
- List all possible truth values for and . Since each variable can be either true (T) or false (F), there are 4 possible combinations.
- Compute the truth values for each sub-expression step by step.
- Determine whether the statement is a tautology by seeing if the final column is always true.
Here is the truth table layout for the expression:
T | T | F | F | F | T |
T | F | F | T |
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Math Problem Analysis
Mathematical Concepts
Propositional Logic
Truth Tables
Logical Implication
Formulas
Negation (~)
Conjunction (∧)
Implication (→)
Theorems
Truth Table Theorem
Suitable Grade Level
Grade 9-12