Problem 29 - Thinking Mathematically

Question

Write the converse, inverse, and contrapositive of each statement. \(\sim q \rightarrow \sim r\)

Step-by-Step Solution

Answer

The converse of the statement \(\sim q \rightarrow \sim r\) is \(\sim r \rightarrow \sim q\), the inverse is \(q \rightarrow r\), and the contrapositive is \(r \rightarrow q\).

Step 1: Converse of the Statement

The converse of any given statement interchanges the hypothesis and the conclusion. Here, the given statement is the form \(\sim q \rightarrow \sim r\), the converse will be \(\sim r \rightarrow \sim q\).

Step 2: Inverse of the Statement

Inverse of a statement is created by negating both the hypothesis and the conclusion. Negating the given statement \(\sim q \rightarrow \sim r\) will result in \(q \rightarrow r\), which is the inverse.

Step 3: Contrapositive of the Statement

Contrapositive of a statement is achieved by interchanging and negating both hypothesis and conclusion. Doing this for the given statement \(\sim q \rightarrow \sim r\), gives the contrapositive as \(r \rightarrow q\).

Key Concepts

Converse

In logic, the converse of a statement switches the roles of the hypothesis and the conclusion. If you have a statement in the form "If A, then B," the converse would be "If B, then A." For our specific example, the given statement is \( \sim q \rightarrow \sim r \). To form the converse, simply interchange \( \sim q \) and \( \sim r \). Thus, the converse becomes \( \sim r \rightarrow \sim q \). This structure helps to see if the reverse condition holds true.
Interchanging the hypothesis and conclusion is a simple yet powerful tool in logical reasoning. Remember, the truth of the converse is not guaranteed by the truth of the original statement.

Inverse

To find the inverse of a logical statement, you must negate both the hypothesis and the conclusion. Given a statement "If A, then B," the inverse would be "If not A, then not B."
With our provided statement \( \sim q \rightarrow \sim r \), forming the inverse means negating both parts. As \( \sim q \) is already a negation, negating it turns \( \sim q \) into \( q \). Similarly, \( \sim r \) becomes \( r \). Hence, the inverse of our statement is \( q \rightarrow r \).
The inverse is useful for exploring different logical relationships, but like the converse, it does not necessarily share the truth value of the original statement.

Contrapositive

The contrapositive of a statement combines the techniques used to form the converse and the inverse. It involves both swapping and negating the hypothesis and conclusion.
For a statement like "If A, then B," the contrapositive would be "If not B, then not A." Applying this to \( \sim q \rightarrow \sim r \), you first interchange \( \sim q \) and \( \sim r \) to get \( \sim r \rightarrow \sim q \). Then negate both, turning it into \( r \rightarrow q \).
An intriguing property of the contrapositive is that it always shares the same truth value as the original statement. Understanding this concept is crucial in logical reasoning.

Hypothesis

In logical statements, the hypothesis is the "if" part, typically representing a condition or scenario. In a statement "If A, then B," A is the hypothesis.
For \( \sim q \rightarrow \sim r \), \( \sim q \) stands as the hypothesis. This is the starting condition for evaluating the truth of the entire statement.
Grasping what the hypothesis represents helps in forming logical conclusions and exploring other logical constructions like converse or contrapositive.

Conclusion

The conclusion is the "then" part of a logical statement, often representing the outcome of the given hypothesis. In the structure "If A, then B," B is the conclusion.
In \( \sim q \rightarrow \sim r \), the conclusion is \( \sim r \). This is what follows if the hypothesis \( \sim q \) holds true.
Understanding the role of the conclusion in logical statements is essential for evaluating the validity and relationships between different logical expressions.