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:

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

Where:

  • ¬p\neg p means "not pp" (negation of pp),
  • \land is the logical AND,
  • \rightarrow is the logical implication.

Steps to Construct the Truth Table:

  1. List all possible truth values for pp and qq. Since each variable can be either true (T) or false (F), there are 4 possible combinations.
  2. Compute the truth values for each sub-expression step by step.
  3. 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:

ppqq¬p\neg pp¬qp \land \neg q¬pq\neg p \land q(¬pq)(p¬q)(\neg p \land q) \rightarrow (p \land \neg q)
TTFFFT
TFFT

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

Negation (~)
Conjunction (∧)
Implication (→)

Theorems

Truth Table Theorem

Suitable Grade Level

Grade 9-12