Math Problem Statement

Write the following two statements in symbolic form and determine whether they are logically equivalent. Include a truth table and explanation.

Solution

Let's break down and solve the problem step by step.

Step 1: Define Simple Statements

Let's denote the following simple statements from the problem:

  • pp: "Butch bought the required calculator for the course."
  • qq: "Butch passed the course."

Step 2: Translate the Statements into Symbolic Logic

  1. First Statement (If Butch did not buy the calculator, then she cannot pass the course):

    • This can be written as a conditional statement:
    • "If not pp, then not qq."
    • Symbolically: ¬p¬q\neg p \rightarrow \neg q
  2. Second Statement (Butch bought the required calculator for the course and she passed the course):

    • This is a conjunction of two statements:
    • "Butch bought the calculator" (which is pp) and "she passed the course" (which is qq).
    • Symbolically: pqp \land q

Step 3: Truth Table

Let's construct a truth table for both statements and compare them.

ppqq¬p\neg p¬q\neg q¬p¬q\neg p \rightarrow \neg qpqp \land q
TTFFTT
TFFTTF
FTTFFF
FFTTTF

Step 4: Explanation of Truth Table

  • For the first statement ¬p¬q\neg p \rightarrow \neg q:
    • The conditional is only false when the antecedent (¬p\neg p) is true, and the consequent (¬q\neg q) is false. In this case, the truth table shows the statement is false when p=Fp = F and q=Tq = T, which matches the logical rule for implications.
  • For the second statement pqp \land q:
    • The conjunction is only true when both pp and qq are true, which occurs only in the first row of the truth table.

Step 5: Are the Statements Logically Equivalent?

  • The statements are not logically equivalent because their truth values differ across the rows in the truth table. Specifically, the first statement ¬p¬q\neg p \rightarrow \neg q is true in the third row, while the second statement pqp \land q is false in that same row.

Conclusion

The two statements are not logically equivalent as their truth tables differ.

Would you like to dive deeper into any part of this explanation or have further questions?

Related Questions:

  1. What does it mean for two statements to be logically equivalent?
  2. How do you construct a truth table for more complex logical statements?
  3. What is the significance of a truth table in determining logical equivalence?
  4. Can conditional statements (\rightarrow) be written in different logical forms?
  5. How does negation affect the truth values in logical statements?

Tip:

In logical reasoning, always start by defining the simple components of a statement clearly before translating them into symbolic form to avoid confusion.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Logical Equivalence
Truth Tables

Formulas

Implication: p → q
Conjunction: p ∧ q
Negation: ¬p

Theorems

Truth Table Method for Logical Equivalence

Suitable Grade Level

Grades 10-12 (High School Level)