Math Problem Statement

Is this table correct

Is this table correct?

Solution

The truth table shown in your image evaluates the logical expression [(pr)(rq)]p[(p \to r) \land (r \land q)] \to p. Let's break down the table step by step and verify the correctness.

Columns:

  1. pp: The first column lists the truth values of pp (True (T) or False (F)).
  2. qq: The second column lists the truth values of qq.
  3. rr: The third column lists the truth values of rr.
  4. pqp \to q: This is the implication from pp to qq.
  5. rqr \land q: The logical AND between rr and qq.
  6. (pr)(rq)(p \to r) \land (r \land q): A combination of the implication prp \to r and rqr \land q.
  7. [(pr)(rq)]p[(p \to r) \land (r \land q)] \to p: The final expression.

Steps for verification:

  1. pqp \to q: The implication is false only when pp is true and qq is false; otherwise, it's true.
  2. rqr \land q: The AND operation is true only if both rr and qq are true.
  3. prp \to r: The implication is false only when pp is true and rr is false.
  4. (pr)(rq)(p \to r) \land (r \land q): This is true only when both parts of the conjunction are true.
  5. [(pr)(rq)]p[(p \to r) \land (r \land q)] \to p: The final column will be false if the premise is true and pp is false; otherwise, it will be true.

After reviewing the image, the table seems consistent with these steps. Therefore, based on a quick visual check, the table appears to be correct.

Would you like a more detailed breakdown of a particular row or column? Feel free to ask any further questions!


Here are 5 questions you might explore based on this:

  1. What happens if you change the logical operators in the formula?
  2. How does the truth table change if you reverse the implication in the expression?
  3. Can we simplify the logical expression to a more concise form?
  4. What are the common logical identities that relate to this type of expression?
  5. How does adding more variables affect the complexity of the truth table?

Tip: When analyzing truth tables, pay attention to implications (\to)—they are false only when the premise is true, but the conclusion is false.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Truth Tables
Logical Implications
Conjunction (AND)

Formulas

Implication (p → q): True unless p is true and q is false
Conjunction (r ∧ q): True when both r and q are true

Theorems

Truth Table Evaluation
Logical Equivalencies

Suitable Grade Level

Grades 11-12, College Level