Math Problem Statement
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:
- : "Butch bought the required calculator for the course."
- : "Butch passed the course."
Step 2: Translate the Statements into Symbolic Logic
-
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 , then not ."
- Symbolically:
-
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 ) and "she passed the course" (which is ).
- Symbolically:
Step 3: Truth Table
Let's construct a truth table for both statements and compare them.
T | T | F | F | T | T |
T | F | F | T | T | F |
F | T | T | F | F | F |
F | F | T | T | T | F |
Step 4: Explanation of Truth Table
- For the first statement :
- The conditional is only false when the antecedent () is true, and the consequent () is false. In this case, the truth table shows the statement is false when and , which matches the logical rule for implications.
- For the second statement :
- The conjunction is only true when both and 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 is true in the third row, while the second statement 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:
- What does it mean for two statements to be logically equivalent?
- How do you construct a truth table for more complex logical statements?
- What is the significance of a truth table in determining logical equivalence?
- Can conditional statements () be written in different logical forms?
- 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)