Problem 10 - Discrete Mathematics with Applications

Question

If Peter is married, he is happy. If he is happy, then he does not read the computer magazine. He does read the computer magazine.

Step-by-Step Solution

Answer

Using contrapositive statements derived from the given conditions, we conclude that since Peter reads the computer magazine and is not happy, \(\therefore\) Peter is unmarried.

Step 1: Analyze the given conditions

We are given the following three statements involving Peter: 1. If Peter is married, he is happy. 2. If he is happy, then he does not read the computer magazine. 3. He does read the computer magazine.

Step 2: Use Contrapositive Rule

The contrapositive rule states that if a statement "If A, then B" is true, then the statement "If not B, then not A" is also true. Let's apply this rule to the given statements: 1. If Peter is married (A), he is happy (B). Contrapositive: If Peter is not happy (not B), then he is not married (not A). 2. If Peter is happy (A), then he does not read the computer magazine (B). Contrapositive: If Peter does read the computer magazine (not B), then he is not happy (not A).

Step 3: Combine the Contrapositive Statements

We are given the statement that Peter reads the computer magazine, so we can now combine the contrapositive statements: - Peter does read the computer magazine (not B). - So, based on the contrapositive of Statement 2: Peter is not happy (not A). - Now, using the contrapositive of Statement 1: Since Peter is not happy (not B), then he is not married (not A).

Step 4: Conclusion

Since we used the given conditions and contrapositive statements, we conclude that \(\therefore\) Peter is unmarried.

Key Concepts

Contrapositive in Logic

The concept of a contrapositive in logic might sound complicated, but it's actually quite straightforward and incredibly useful. In logic, when we have a conditional statement of the form "if A, then B," we can create a contrapositive statement. The contrapositive of "if A, then B" is "if not B, then not A." Simply put, we swap and negate both parts of the statement.

Here's why this is helpful: if the original statement is true, the contrapositive will also be true. This rule provides a new way to look at problems and can help us make deductions.

• Original: If A, then B
• Contrapositive: If not B, then not A

Consider this in our exercise with Peter. We take "If Peter is married, he is happy" and its contrapositive becomes "If Peter is not happy, he is not married." By understanding and applying contrapositives, we can effectively draw conclusions from existing information.

Logical Deductions

Logical deductions allow us to derive a conclusion from given statements known to be true. This often involves using rules of logic to combine or modify the statements we have. In our exercise, we first analyzed the statements by Peter:

• He is married, so he is happy (conditional statement).
• If happy, he doesn’t read magazines (conditional statement).
• He does read magazines (given fact).

The strategy with deductions is akin to a puzzle. We used the contrapositive logic to translate our given facts into the contrapositive inferences. Through logical deductions, we ended up with the conclusion that Peter is not married.

By combining these statements, recognizing which are contrapositive of the given facts, we form a logical chain that leads us to conclude Peter’s marital status definitively.

Conditional Statements

In logic, a conditional statement is a compound sentence that links an antecedent ("if" part) to a consequent ("then" part). These statements are fundamental in logical reasoning and constructing arguments, as they suggest a cause-and-effect relationship between the two linked propositions.

In the example involving Peter, we had a series of conditional statements:

• If Peter is married, then he is happy.
• If Peter is happy, then he does not read the computer magazine.

These conditionals provided a framework for determining an outcome. By using these conditional statements along with their contrapositives, we processed the given facts and asserted logical conclusions.

Understanding conditional statements is crucial for anyone studying logic, as they help deduce the possible outcomes based on a set of given conditions. They act as building blocks for creating logical sequences and making sound rational conclusions in problem-solving scenarios.