Question

For each of the following statements, find the corresponding inverse, converse, and contrapositive. (a) If the stars are shining, then it is the middle of the night. (b) If the Wizards won, then they scored at least 100 points. (c) If the exam is hard, then the highest grade is less than 90 .

Short answer

Inverse: Negate both parts; Converse: Switch parts; Contrapositive: Switch and negate both parts.

Step-by-step solution

Understand Conditional Statements

A conditional statement has the form 'if P, then Q', where P is the hypothesis and Q is the conclusion. We identify the parts: (a) P: 'the stars are shining', Q: 'it is the middle of the night'; (b) P: 'the Wizards won', Q: 'they scored at least 100 points'; (c) P: 'the exam is hard', Q: 'the highest grade is less than 90'.

Formulate Inverse Statements

To form the inverse, negate both the hypothesis and the conclusion. (a) If the stars are not shining, then it is not the middle of the night. (b) If the Wizards did not win, then they scored less than 100 points. (c) If the exam is not hard, then the highest grade is at least 90.

Formulate Converse Statements

To form the converse, switch the hypothesis and the conclusion. (a) If it is the middle of the night, then the stars are shining. (b) If they scored at least 100 points, then the Wizards won. (c) If the highest grade is less than 90, then the exam is hard.

Formulate Contrapositive Statements

To form the contrapositive, switch and negate both the hypothesis and the conclusion. (a) If it is not the middle of the night, then the stars are not shining. (b) If they scored less than 100 points, then the Wizards did not win. (c) If the highest grade is at least 90, then the exam is not hard.

Key concepts

Inverse Statement

An inverse statement is derived from a conditional statement by negating both the hypothesis and the conclusion. Let's break down what this implies. If your original conditional statement is "If P, then Q," the inverse becomes "If not P, then not Q." For example, let's examine the statement "If the stars are shining, then it is the middle of the night." The inverse of this would be: "If the stars are not shining, then it is not the middle of the night." Here, we're challenging both parts of the original idea.

Notice how simple it is to form the inverse by just adding "not" before both parts. It's crucial to understand that the truth value of the inverse isn't always the same as the original statement. They are logically independent. While creating inverse statements, always ensure you carefully choose your words for clarity.

Converse Statement

Creating a converse statement involves switching the hypothesis and the conclusion of the original conditional statement. For a statement like "If P, then Q," the converse is "If Q, then P." This rearrangement often changes the meaning quite dramatically.

Let's take the example: "If the exam is hard, then the highest grade is less than 90." The converse here becomes "If the highest grade is less than 90, then the exam is hard."

It's critical to recognize that a converse statement may have a different truth value than its original statement. Just because a conditional statement is true doesn't automatically make its converse true. Always analyze each statement in its context to verify its validity.

• The converse switches hypothesis and conclusion.
• The truth value is often different from the original.

Contrapositive Statement

The contrapositive statement combines elements from both the inverse and the converse. You not only switch the hypothesis and conclusion but also negate both. If "If P, then Q" is your original, the contrapositive looks like "If not Q, then not P." Changing both elements actually maintains the truth value from the original in many cases.

Consider this example: "If the Wizards won, then they scored at least 100 points." The contrapositive would be: "If they scored less than 100 points, then the Wizards did not win."

An important property of the contrapositive is that it shares the same truth value as the original statement. This logical connection can be very powerful in reasoning and problem-solving.

• Switch and negate both parts.
• Retains the same truth value as the original.

Logical Reasoning

Logical reasoning involves drawing conclusions based on given statements or premises. This can be guided by constructing conditional, inverse, converse, and contrapositive statements. Each type of statement serves a different purpose in logic, allowing us to better clarify assumptions and conclusions.

When engaging in logical reasoning, it's essential to consider the relationships between statements. How does altering one statement affect its truth? Does switching or negating make the statement lose its validity? These are the kinds of questions that propel critical thinking.

Logical reasoning helps in:

• Evaluating complex problems step-by-step.
• Identifying faulty assumptions.
• Constructing well-supported arguments.

Using these tools effectively can help in many areas, from mathematical proofs to everyday decision-making. Always question and analyze the reasoning paths you choose.