Question

Critical Thinking The inverse of a conditional is formed by negating both the hypothesis and conclusion of the conditional. Conditional: If it is raining, then it is cloudy. Inverse: \(\quad\) If it is not raining, then it is not cloudy. The contrapositive of a conditional is formed by negating both the hypothesis and conclusion of the converse of the conditional. Converse: If it is cloudy, then it is raining. Contrapositive: If it is not cloudy, then it is not raining. a. Write the inverse and contrapositive of this statement. If a figure has five sides, then it is a pentagon. b. Write a conditional. Then write its converse, inverse, and contrapositive.

Short answer

Inverse: "If a figure doesn't have five sides, it's not a pentagon." Contrapositive: "If it's not a pentagon, it doesn't have five sides."

Step-by-step solution

Understanding the Problem

We need to find the inverse and contrapositive of a given conditional statement. We also need to create our own conditional statement and find its converse, inverse, and contrapositive.

Part a - Formulating the Inverse

The inverse of a conditional statement is formed by negating both its hypothesis and conclusion. For the statement "If a figure has five sides, then it is a pentagon," the hypothesis is "a figure has five sides," and the conclusion is "it is a pentagon." The inverse would be: "If a figure does not have five sides, then it is not a pentagon."

Part a - Formulating the Contrapositive

The contrapositive of a conditional is formed by negating both the hypothesis and conclusion of the converse of the statement. First, we find the converse: "If it is a pentagon, then a figure has five sides." Then we negate both parts to form the contrapositive: "If it is not a pentagon, then a figure does not have five sides."

Create a Conditional Statement for Part b

Let's create the conditional statement "If a student studies, then they pass the test." Here, the hypothesis is "a student studies," and the conclusion is "they pass the test."

Part b - Formulating the Converse

The converse is formed by swapping the hypothesis and conclusion of the original statement. For "If a student studies, then they pass the test," the converse is: "If they pass the test, then a student studies."

Part b - Formulating the Inverse

Negate both the hypothesis and the conclusion of the original statement. The inverse becomes: "If a student does not study, then they do not pass the test."

Part b - Formulating the Contrapositive

Find the converse, then negate both the hypothesis and conclusion. The contrapositive of the original statement is: "If they do not pass the test, then a student does not study."

Key concepts

Inverse

The inverse of a conditional statement is a logical operation that involves negating both the hypothesis and conclusion of the original statement. It transforms a statement like "If it is raining, then it is cloudy" into "If it is not raining, then it is not cloudy." This transformation changes the original conditions.
While the inverse may not always be true if the original statement is true, it provides another perspective on the relationship between the conditions in the hypothesis and conclusion.
To formulate the inverse, simply:

• Identify the hypothesis and conclusion from the original statement.
• Negate both parts.

For example, consider "If a figure has five sides, then it is a pentagon." The hypothesis, "a figure has five sides," and the conclusion, "it is a pentagon," are negated to construct the new statement: "If a figure does not have five sides, then it is not a pentagon."
This process helps in exploring all logical possibilities of a statement, making it a valuable tool in logic and critical thinking.

Contrapositive

The contrapositive of a conditional statement is a transformation achieved by negating both the hypothesis and conclusion of the converse of the original statement. This can sometimes make the logical relationships clearer.
For example, starting with "If it is raining, then it is cloudy," the converse is "If it is cloudy, then it is raining." The contrapositive then becomes "If it is not cloudy, then it is not raining."
Interestingly, the truth values of the original conditional statement and its contrapositive are the same. Therefore, proving the contrapositive is often used in mathematical proofs. To formulate a contrapositive, follow these steps:

• Form the converse by swapping the hypothesis and conclusion.
• Negate both the hypothesis and conclusion of the converse.

Using the statement "If a figure has five sides, then it is a pentagon," the converse is "If it is a pentagon, then it is a figure with five sides," which becomes "If it is not a pentagon, then it is not a figure with five sides" when negated to make the contrapositive. Understanding the contrapositive aids significantly in logical deductions and reasoning.

Hypothesis and Conclusion Negation

In logic, understanding how to work with the hypothesis and conclusion is key to forming different types of statements like inverse and contrapositive. Negation plays a vital role here.
Negating a hypothesis or conclusion means stating the opposite. For instance, if the hypothesis is "A student studies," negation would be "A student does not study." Such negations change the truth values of the statements, offering new perspectives.

• The hypothesis is the "if" part of a conditional statement.
• The conclusion is the "then" part of a conditional statement.

Negating both components transforms the statement's logical framework. Take the statement "If it is sunny, then it is warm." Negation of the hypothesis becomes "If it is not sunny," and negation of the conclusion becomes "then it is not warm."
Thus, when you negate the hypothesis and conclusion, you formulate the inverse. Moreover, this negation approach is crucial in creating meaningful variations such as the contrapositive, equipping you with robust tools for logical arguments and critical thinking. Understanding negation deepens comprehension of logical equivalents and assists in problem-solving across disciplines.