Math Problem Statement
Help me answer the question that
Construct a truth table for the statement (~p ∧ q) → (p ∧ ~q). Then determine if the statement is a tautology.
Solution
The statement in the image requires constructing a truth table for the logical expression:
To determine whether the statement is a tautology, we need to construct the truth table for all possible values of and , evaluate the truth of the expression for each case, and then see if the statement is always true.
Steps:
- List all possible truth values for and .
- Evaluate , , , and .
- Use the implication () to find the truth value of the entire expression.
I'll begin by constructing the truth table now.Here is the completed truth table for the expression :
\hline p & q & \neg p & \neg q & \neg p \land q & p \land \neg q & (\neg p \land q) \rightarrow (p \land \neg q) \\ \hline T & T & F & F & F & F & T \\ T & F & F & T & F & T & T \\ F & T & T & F & T & F & F \\ F & F & T & T & F & F & T \\ \hline \end{array}$$ ### Conclusion: The expression is **not** a tautology because in one case (when $$p = \text{False}$$ and $$q = \text{True}$$), the statement is **false**. A tautology must be true in all cases. Do you need further details or explanations? Here are 5 questions to expand on this: 1. What is the difference between a tautology and a contradiction in logic? 2. How is the implication $$p \rightarrow q$$ defined in logic? 3. Can a logical expression be simplified using logical equivalences? 4. How do truth tables help in validating logical arguments? 5. What are some common logical operators used in propositional logic? **Tip**: When constructing truth tables, always start by evaluating individual components like negations and conjunctions before tackling implications.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Propositional Logic
Truth Tables
Logical Implication
Formulas
Truth table for (~p ∧ q) → (p ∧ ~q)
Theorems
Tautology in logic
Suitable Grade Level
College-Level Logic/Discrete Mathematics