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:

(¬pq)(p¬q)(\neg p \land q) \rightarrow (p \land \neg q)

To determine whether the statement is a tautology, we need to construct the truth table for all possible values of pp and qq, evaluate the truth of the expression for each case, and then see if the statement is always true.

Steps:

  1. List all possible truth values for pp and qq.
  2. Evaluate ¬p\neg p, ¬q\neg q, ¬pq\neg p \land q, and p¬qp \land \neg q.
  3. Use the implication (\rightarrow) 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 (¬pq)(p¬q)(\neg p \land q) \rightarrow (p \land \neg q):

\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

By Image

Drop file here or Click Here to upload

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