Problem 30 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The converse of \(\sim p \rightarrow r\) is \( r \rightarrow \sim p\). The inverse is \( p \rightarrow \sim r\). The contrapositive is \(\sim r \rightarrow p\).

Step 1: Find the Converse

The converse of a conditional statement switches the hypothesis and conclusion. Therefore, the converse of \(\sim p \rightarrow r\) is \( r \rightarrow \sim p\).

Step 2: Find the Inverse

The inverse of a conditional statement negates both the hypothesis and conclusion of the original statement. The negation of \(\sim p\) is \(p\) and the negation of \(r\) is \(\sim r\). Therefore, the inverse of \(\sim p \rightarrow r\) is \( p \rightarrow \sim r\).

Step 3: Find the Contrapositive

The contrapositive of a statement switches the hypothesis and conclusion and then negates both. Thus, the contrapositive of \(\sim p \rightarrow r\) is \(\sim r \rightarrow p\).

Key Concepts

Understanding Conditional Statements

Conditional statements are fundamental to logical reasoning and writing. They help us relate two statements in a logical way. A conditional statement is often expressed in the form "If...then...". For example, "If it rains, then the ground will be wet." The statement can be represented in logical notation as \( p \rightarrow q \), where \( p \) is the hypothesis (or condition) and \( q \) is the conclusion.Conditional statements are pivotal in mathematics, computer science, and everyday problem-solving. They allow us to determine what conclusions can be drawn if certain conditions are met. Recognizing when these conditions are met will enable you to identify the resulting consequences.

Exploring Converse Statements

In logic, the converse of a conditional statement is formed by swapping its hypothesis and conclusion. This means that if the original statement is \( p \rightarrow q \), then the converse would be \( q \rightarrow p \).However, it's important to note that the truth value of the converse is independent of the original statement. Just because the original is true, doesn't necessarily mean its converse is true too.

• Example: Original: "If it rains, then the ground is wet."
• Converse: "If the ground is wet, then it rains."

In our exercise, the converse changes \( \sim p \rightarrow r \) to \( r \rightarrow \sim p \), switching the order of hypothesis and conclusion.

Identifying Inverse Statements

The inverse of a conditional statement retains the order of elements but negates both the hypothesis and conclusion. For a statement \( p \rightarrow q \), the inverse would be \( \sim p \rightarrow \sim q \).Negating a statement involves turning it into its logical opposite. For instance, 'p' becomes 'not p', denoted as \( \sim p \).

• Example: Original: "If it rains, then the ground is wet."
• Inverse: "If it does not rain, then the ground is not wet."

In our exercise, the inverse negates both terms in \( \sim p \rightarrow r \), resulting in \( p \rightarrow \sim r \).

Discovering Contrapositive Statements

The contrapositive of a conditional statement swaps and negates both the hypothesis and conclusion. For the statement \( p \rightarrow q \), the contrapositive is \( \sim q \rightarrow \sim p \).Interestingly, a statement and its contrapositive always share the same truth value. If the original statement is true, so is its contrapositive, and vice versa. This makes contrapositive statements a reliable method of logical proof.

• Example: Original: "If it rains, then the ground is wet."
• Contrapositive: "If the ground is not wet, then it does not rain."

In our exercise, the contrapositive of \( \sim p \rightarrow r \) becomes \( \sim r \rightarrow p \), both swapping and negating the original terms.