Problem 32 - Thinking Mathematically

Question

Express each statement in "if ... then" form. (More than one correct wording in "if... then" form may be possible.) Then write the statement's converse, inverse, and contrapositive. All senators are politicians.

Step-by-Step Solution

Answer

If-Then Form: 'If someone is a senator, then they are a politician.'; Converse: 'If someone is a politician, then they are a senator.'; Inverse: 'If someone is not a senator, then they are not a politician.'; Contrapositive: 'If someone is not a politician, then they are not a senator.'

Step 1: Conversion to If-Then Form

The given statement 'All senators are politicians.' can be converted into if-then form as: 'If someone is a senator, then they are a politician.'

Step 2: Finding the Converse

The converse of a statement switches the hypothesis and the conclusion of the original If-Then Statement. So, the converse of the statement is: 'If someone is a politician, then they are a senator.'

Step 3: Finding the Inverse

The inverse of a statement negates both the hypothesis and the conclusion of the original if-then Statement. So, the inverse of the statement is: 'If someone is not a senator, then they are not a politician.'

Step 4: Finding the Contrapositive

The contrapositive of a statement switches the hypothesis and the conclusion of the original if-then statement and then negates them. So, the contrapositive of the statement is: 'If someone is not a politician, then they are not a senator.'

Key Concepts

If-Then Statements

Using if-then statements is a fundamental concept in mathematical logic. An if-then statement is a logical structure that connects two statements: a hypothesis and a conclusion. It's often written in the form "If P, then Q," where P is the hypothesis and Q is the conclusion.

In the provided exercise, we converted the statement "All senators are politicians" into an if-then form, resulting in: "If someone is a senator, then they are a politician." This means that being a senator guarantees that someone is a politician.

• The part after "if" is called the hypothesis ("someone is a senator").
• The part following "then" is the conclusion ("they are a politician").

The statement assumes that the hypothesis naturally leads to the conclusion, establishing a cause-and-effect-like relationship.

Converse Statement

A converse statement flips the hypothesis and conclusion of an if-then statement. This transformation can drastically change the meaning of the original statement.

To find the converse of "If someone is a senator, then they are a politician," we switch the two parts: "If someone is a politician, then they are a senator." Notice how this new statement suggests that every politician must be a senator, which is quite different from the original statement's meaning.

• Original: "If someone is a senator, then they are a politician."
• Converse: "If someone is a politician, then they are a senator."

It's important to note that a converse statement is not always logically equivalent to the original. Just because one statement is true does not guarantee the truth of its converse.

Inverse Statement

The inverse of an if-then statement involves negating both the hypothesis and the conclusion.

Let's apply this to our example statement, "If someone is a senator, then they are a politician."
By making both parts negative, we get: "If someone is not a senator, then they are not a politician." This new statement suggests that not being a senator automatically means not being a politician, which is a fundamentally different claim from the original.

• Original: "If someone is a senator, then they are a politician."
• Inverse: "If someone is not a senator, then they are not a politician."

Like with the converse, the truth of the inverse is independent of the original statement. An inverse statement may or may not be true, regardless of the original statement's truth.

Contrapositive Statement

The contrapositive of an if-then statement is a powerful logical transformation. It swaps and negates both the hypothesis and the conclusion. In simple terms, it combines the processes used to create the converse and the inverse.

For the statement "If someone is a senator, then they are a politician," the contrapositive is: "If someone is not a politician, then they are not a senator." This transformation is unique because the contrapositive of a true statement is always true, and similarly, the contrapositive of a false statement is always false.

• Original: "If someone is a senator, then they are a politician."
• Contrapositive: "If someone is not a politician, then they are not a senator."

This logical equivalence makes the contrapositive extremely useful in proofs and reasoning within mathematics. When proving the truth of a statement, one often proves the truth of its contrapositive.