Problem 39 - Thinking Mathematically

Question

Determine which, if any, of the three given statements are equivalent. You may use information about a conditional statement's converse, inverse, or contrapositive, De Morgan's laws, or truth tables. a. If he is guilty, then he does not take a lie-detector test. b. He is not guilty or he takes a lie-detector test. c. If he is not guilty, then he takes a lie-detector test.

Step-by-Step Solution

Answer

Statement a and Statement b are equivalent. Statement c is not equivalent to the other two.

Step 1: Translate Statements into Logical Format

Translate the statements into logical format. 'If...then...' statements can be translated into \(P \Rightarrow Q\) where P is the premise and Q is the conclusion. Using this, translate the following statements: a. If he is guilty (P), then he does not take a lie-detector test (~Q). This is \(P \Rightarrow \sim Q\). b. He is not guilty (~P) or he takes a lie-detector test (Q). This is \( \sim P \vee Q\). c. If he is not guilty (~P), then he takes a lie-detector test (Q). This is \(\sim P \Rightarrow Q\).

Step 2: Use Logical Equivalences

Use De Morgan's laws and the logical equivalences of conditionals. The conditional \(P \Rightarrow Q\) is logically equivalent to \(\sim P \vee Q\) (not P or Q). Checking the three statements, it's clear that Statement a and Statement b are logically equivalent because \(P \Rightarrow \sim Q\) is the same as \(\sim P \vee Q\). However, Statement c is not equivalent to Statement a or Statement b.

Step 3: Draw a Truth Table (Optional)

An optional step for further clarity would be to draw a truth table. This would systematically check all possible truth values for P and Q, and compare the results for the logical expressions of each of the three statement. This will confirm that indeed, Statement a and b are equivalent while Statement c is not.

Key Concepts

Conditional Statements

Understanding conditional statements is crucial in both mathematics and computer science. A conditional statement, also known as an 'implication', takes the form 'If P, then Q', where P is called the 'antecedent' and Q is the 'consequent'. This can be expressed in logical terms as a logical implication, represented by the symbol \rightarrow., pronounced as 'P implies Q'.

In logical notation, this implication is written as \(P \Rightarrow Q\). The truth of a conditional statement is determined by whether or not the implication is logically consistent. For example, if 'P' is true and 'Q' is also true, then the implication holds. But if 'P' is true and 'Q' is false, the implication does not hold, and the entire statement is considered false. The remaining cases, where 'P' is false (regardless of the truth value of 'Q'), the implication is always considered to be true.

This might seem counterintuitive at first, but in logic, an implication is only false when a true antecedent leads to a false consequent. As part of exploring logical equivalences, we come to understand that a conditional statement can be expressed in different forms, where its converse, inverse, and contrapositive each have distinct meanings and truth values.

De Morgan's Laws

De Morgan's laws are pivotal in the study of logic, as they provide a way to transform complex logical expressions. These laws describe how the negation of conjunctions and disjunctions behave and are often used in computer science, mathematics, and logic to simplify statements or algorithms. The two laws are stated as:

• The negation of a conjunction is equivalent to the disjunction of the negations. Mathematically, this is represented as \(\sim(P \land Q) \equiv \sim P \vee \sim Q\).
• The negation of a disjunction is equivalent to the conjunction of the negations. Expressed in symbols: \(\sim(P \vee Q) \equiv \sim P \land \sim Q\).

These laws can be applied to our exercise by transforming the conditional statements into disjunctions. For instance, the statement 'If P, then Q' is logically equivalent to '\(~P \vee Q\)', allowing us to compare the equivalency of different forms of statements by examining their components under negation and disjunction.

Truth Tables

Truth tables are an essential tool for illustrating and analyzing the truth values of logical expressions. They systematically list all possible combinations of truth values for given propositions and show the result of logical operations for each combination. By filling out a truth table, you can determine the validity or equivalence of logical sentences in a clear and organized manner.

A truth table has a column for each of the variables (e.g., P, Q) and a column for the compound statements (like \(P \Rightarrow Q\)). For simplicity, with two variables, there are four possible combinations of truth values: both can be true, both can be false, or one can be true while the other is false.

By comparing the columns of truth values, we can decide whether different logical expressions are equivalent. In our exercise, a truth table could confirm that the statements 'If P, then Q' (\(P \Rightarrow Q\)) and 'Not P or Q' (\(\sim P \vee Q\)) are equivalent, a concept central to understanding conditional statements and their logical implications.