Problem 66 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write the converse and the contrapositive to the following statements. (a) If the measure of angle \(A B C\) is \(45^{\circ}\), then angle \(A B C\) is an acute angle. (b) If \(a

Step-by-Step Solution

Answer

(a) Converse: 'If angle ABC is acute, then it's 45°.' Contrapositive: 'If angle ABC isn't acute, then it's not 45°.' (b) Converse: 'If a² < b², then a < b.' Contrapositive: 'If a² ≥ b², then a ≥ b.'

Step 1: Understanding the Original Statement (a)

The original statement is: "If the measure of angle \(ABC\) is \(45^{\circ}\), then angle \(ABC\) is an acute angle." This is a conditional statement of the form 'If \(P\), then \(Q\).' Here, \(P\) is 'the measure of angle \(ABC\) is \(45^{\circ}\),' and \(Q\) is 'angle \(ABC\) is an acute angle.'

Step 2: Writing the Converse (a)

The converse of a conditional statement switches the hypothesis and conclusion. Thus, the converse of the statement is: "If angle \(ABC\) is an acute angle, then the measure of angle \(ABC\) is \(45^{\circ}\)."

Step 3: Writing the Contrapositive (a)

The contrapositive of a conditional statement negates both the hypothesis and the conclusion and switches them. The contrapositive is: "If angle \(ABC\) is not an acute angle, then the measure of angle \(ABC\) is not \(45^{\circ}\)."

Step 4: Understanding the Original Statement (b)

The original statement is: "If \(a

Step 5: Writing the Converse (b)

The converse of this statement is: "If \(a^{2} < b^{2}\), then \(a < b\)." This simply switches the hypothesis and the conclusion.

Step 6: Writing the Contrapositive (b)

The contrapositive negates both the hypothesis and conclusion and switches them: "If \(a^{2}\geq b^{2}\), then \(a\geq b\)."

Key Concepts

Conditional Statements

In mathematics, a conditional statement is a logical statement that has two parts: a hypothesis and a conclusion. You can think of it as an 'if-then' statement. For example, the statement 'If the measure of angle \(ABC\) is \(45^{\circ}\), then angle \(ABC\) is an acute angle' fits this format perfectly.
Here, "the measure of angle \(ABC\) is \(45^{\circ}\)" acts as the hypothesis (often denoted as \(P\)), and "angle \(ABC\) is an acute angle" is the conclusion (denoted as \(Q\)). Conditional statements can be true or false, depending on the relationship between \(P\) and \(Q\).

• They often take the logical form "If \(P\), then \(Q\)".
• In symbols, this is represented as \(P \rightarrow Q\).

Understanding these statements is fundamental for mathematical reasoning as they reveal how different mathematical facts are interrelated.

Converse Statements

A converse statement is derived from a conditional statement by swapping its hypothesis and conclusion. This means if the original conditional is "If \(P\), then \(Q\)", the converse will be "If \(Q\), then \(P\)". Let's see this applied to an example.
Given a statement: 'If angle \(ABC\) is \(45^{\circ}\), then angle \(ABC\) is an acute angle', the converse would be 'If angle \(ABC\) is an acute angle, then angle \(ABC\) is \(45^{\circ}\)'.

• Writing the converse sometimes changes the truth value of the statement.
• It's crucial to check the validity of the converse independently from the original condition.

Converse statements are important in proofs and problem-solving scenarios where reversing a known condition can lead to new insights.

Contrapositive Statements

The contrapositive of a conditional statement is a bit more intricate compared to finding its converse. To construct a contrapositive, you must both negate and switch the hypothesis and conclusion of the original conditional statement. The pattern goes like this: for "If \(P\), then \(Q\)", the contrapositive is "If not \(Q\), then not \(P\)".
For example, given the statement 'If \(a

This transformation is powerful because the truth of the contrapositive is always linked to the original statement: if the original conditional is true, the contrapositive must also be true, and vice versa.

The contrapositive maintains the truth value of the original statement.

Understanding contrapositive statements can help clarify relationships and is extensively used within mathematical proofs to demonstrate theorems and propositions more robustly.

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Problem 66

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Problem 67

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Problem 68

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Key Concepts

Conditional Statements

In mathematics, a conditional statement is a logical statement that has two parts: a hypothesis and a conclusion. You can think of it as an 'if-then' statement. For example, the statement 'If the measure of angle \(ABC\) is \(45^{\circ}\), then angle \(ABC\) is an acute angle' fits this format perfectly.
Here, "the measure of angle \(ABC\) is \(45^{\circ}\)" acts as the hypothesis (often denoted as \(P\)), and "angle \(ABC\) is an acute angle" is the conclusion (denoted as \(Q\)). Conditional statements can be true or false, depending on the relationship between \(P\) and \(Q\).

• They often take the logical form "If \(P\), then \(Q\)".
• In symbols, this is represented as \(P \rightarrow Q\).

Understanding these statements is fundamental for mathematical reasoning as they reveal how different mathematical facts are interrelated.

Converse Statements

A converse statement is derived from a conditional statement by swapping its hypothesis and conclusion. This means if the original conditional is "If \(P\), then \(Q\)", the converse will be "If \(Q\), then \(P\)". Let's see this applied to an example.
Given a statement: 'If angle \(ABC\) is \(45^{\circ}\), then angle \(ABC\) is an acute angle', the converse would be 'If angle \(ABC\) is an acute angle, then angle \(ABC\) is \(45^{\circ}\)'.

• Writing the converse sometimes changes the truth value of the statement.
• It's crucial to check the validity of the converse independently from the original condition.

Converse statements are important in proofs and problem-solving scenarios where reversing a known condition can lead to new insights.

Contrapositive Statements

The contrapositive of a conditional statement is a bit more intricate compared to finding its converse. To construct a contrapositive, you must both negate and switch the hypothesis and conclusion of the original conditional statement. The pattern goes like this: for "If \(P\), then \(Q\)", the contrapositive is "If not \(Q\), then not \(P\)".
For example, given the statement 'If \(a

This transformation is powerful because the truth of the contrapositive is always linked to the original statement: if the original conditional is true, the contrapositive must also be true, and vice versa.

The contrapositive maintains the truth value of the original statement.

Understanding contrapositive statements can help clarify relationships and is extensively used within mathematical proofs to demonstrate theorems and propositions more robustly.