Question

Write the converse and contrapositive of each statement. a. If the hose is \(60 \mathrm{ft}\) long, then the hose will reach the tomatoes. b. George goes for a walk only if Mary goes for a walk. c. Pamela recites a poem if Andre asked for a poem.

Short answer

Converse of (a): 'If the hose reaches the tomatoes, then it is 60 ft long.' Contrapositive: 'If the hose does not reach the tomatoes, then it's not 60 ft long.' More examples follow similar logic.

Step-by-step solution

Understand Key Terminology

There are three important terms to understand in the context of the exercise: 'inverse,' 'converse,' and 'contrapositive'. The 'inverse' of a statement negates both the hypothesis and the conclusion. The 'converse' switches the hypothesis and conclusion. The 'contrapositive' switches and negates both the hypothesis and conclusion.

Identify Components of Each Statement

Break down each statement into its hypothesis and conclusion. For statement (a), the hypothesis is 'the hose is 60 ft long,' and the conclusion is 'the hose will reach the tomatoes.' For statement (b), the hypothesis is 'Mary goes for a walk,' and the conclusion is 'George goes for a walk.' For statement (c), the hypothesis is 'Andre asked for a poem,' and the conclusion is 'Pamela recites a poem.'

Write the Converse for Each Statement

For statement (a): The converse is 'If the hose reaches the tomatoes, then the hose is 60 ft long.' For statement (b): The converse is 'If George goes for a walk, then Mary goes for a walk.' For statement (c): The converse is 'If Pamela recites a poem, then Andre asked for a poem.'

Write the Contrapositive for Each Statement

For statement (a): The contrapositive is 'If the hose does not reach the tomatoes, then it is not 60 ft long.' For statement (b): The contrapositive is 'If Mary does not go for a walk, then George does not go for a walk.' For statement (c): The contrapositive is 'If Andre did not ask for a poem, then Pamela does not recite a poem.'

Key concepts

Converse

In logic, the **converse** of a statement is created by reversing its original hypothesis and conclusion. If your original statement is structured as "If P, then Q," its converse will be "If Q, then P". This can be very useful, but it's important to remember that the converse of a statement is not necessarily true, even if the original statement is. For example: - Let's take the statement "If the hose is 60 ft long, then the hose will reach the tomatoes." The converse of this statement is "If the hose reaches the tomatoes, then it is 60 ft long." - Notice how the hypothesis and conclusion have switched places.
This change in position can alter the truth value of the statement, so it's crucial to verify these statements separately.

Sometimes, in mathematics and everyday logic, both a statement and its converse can be true. In that case, they form a biconditional statement and can be expressed as "P if and only if Q." But always make sure to check the validity of both the original statement and its converse independently.

Contrapositive

The **contrapositive** is a transformation of a logical statement that involves both switching the hypothesis and conclusion and negating them. When the original statement is formatted as "If P, then Q," the contrapositive becomes "If not Q, then not P." A fascinating aspect of contrapositive statements is that they always share the same truth value as the original statement. This means if the original statement is true, its contrapositive will also be true, and vice versa. Let's examine an example from our exercise: - For the statement "If George goes for a walk only if Mary goes for a walk," the contrapositive will be "If Mary does not go for a walk, then George does not go for a walk." - This process involves two key steps: reversing the positions of the hypothesis and conclusion and adding the 'not' form to each.
These two steps make the contrapositive a powerful tool in logical reasoning, enabling you to infer new information based on an initial statement.

By mastering the contrapositive, you can often simplify complex logical arguments and make deductive reasoning more efficient.

Logical Statements

**Logical statements** are central to both philosophical logic and mathematical reasoning. They form the basis for constructing arguments and determining validity. A logical statement, often called a proposition, is a declarative sentence that is either true or false, but not both. For mathematical purposes, logical statements often come in - 'if-then' structures, such as "If P, then Q," where P and Q are simple propositions.
Understanding this form helps in deriving other related forms, like converse and contrapositive.

The power of logical statements lies in their clarity. By setting up clear conditions for truth, they allow us to explore consequences, assumptions, and conclusions in a structured way. With logical statements, we can follow through intricate chains of reasoning, support problem-solving processes, and make sound arguments. Practicing writing and manipulating logical statements is a fundamental skill in math and everyday reasoning.
It ensures clarity and precision in communication of ideas and can help identify what exactly needs to be proved or addressed.