Problem 63 - Calculus Early Transcendentals: Pearson New International Edition

Question

63\. Write the converse and the contrapositive to the following statements. (a) If it rains today, then I will stay home from work. (b) If the candidate meets all the qualifications, then she will be hired.

Step-by-Step Solution

Answer

Converse (a): If I stay home, it rains. Contrapositive (a): If I don't stay home, it doesn't rain. Converse (b): If hired, she meets qualifications. Contrapositive (b): If not qualified, not hired.

Step 1: Understand the Original Statement

Identify the hypothesis and conclusion in each statement. (a) Hypothesis (If part): It rains today. Conclusion (Then part): I will stay home from work. (b) Hypothesis: The candidate meets all the qualifications. Conclusion: She will be hired.

Step 2: Write the Converse Statements

The converse of a statement is formed by swapping the hypothesis and the conclusion. (a) Converse: If I stay home from work, then it rains today. (b) Converse: If she is hired, then the candidate meets all the qualifications.

Step 3: Write the Contrapositive Statements

The contrapositive of a statement negates both the hypothesis and the conclusion of the converse statement. (a) Contrapositive: If I do not stay home from work, then it does not rain today. (b) Contrapositive: If the candidate does not meet all the qualifications, then she will not be hired.

Key Concepts

Converse Statements

A converse statement is created by flipping the hypothesis and the conclusion of a conditional statement. Imagine a seesaw where you swap the sides; that's what happens with a converse. It's important to note that the truth value of the converse isn't guaranteed to be the same as the original statement. For example, if the original statement is, "If it rains today, then I will stay home from work," the converse would be, "If I stay home from work, then it rains today." In real-life situations, the reason you stay home might not always be rain. That's why converse statements don’t necessarily retain the truth of the original. To gain mastery over forming converse statements, practice by identifying the hypothesis and the conclusion in everyday situations. Swap them to build your own converse statements.

Contrapositive Statements

Contrapositive statements twist both the hypothesis and conclusion, similar to turning them inside out and upside down. Creating a contrapositive involves swapping the hypothesis and conclusion like in a converse, then negating both parts. This might sound a bit complicated, but let's break it down with an example. Take the statement: "If it rains today, then I will stay home from work." The contrapositive is: "If I do not stay home from work, then it does not rain today." Interestingly, unlike the converse, the contrapositive shares the same truth value as the original statement. This means if the original statement is true, the contrapositive is also true. Understanding this relationship is a powerful tool in logical reasoning. Using this pattern, you can strengthen your skills in logic by checking the validity of complex statements in mathematics and real-world contexts.

Conditional Statements

Conditional statements form the backbone of logical reasoning, often recognized by the structure "If... then...". These statements consist of two parts: the hypothesis and the conclusion. Basically, they're promises or predictions, saying that if the first part (the hypothesis) occurs, then the second part (the conclusion) follows. Consider the statement: "If the candidate meets all the qualifications, then she will be hired." Here, "the candidate meets all the qualifications" is the hypothesis, and "she will be hired" is the conclusion. It's essential to remember that conditional statements can be true or false based on the real-world scenario or the logical framework. Oftentimes in logic puzzles or mathematics, we analyze the conditions to validate the truth of these statements. To become proficient, practice by formulating your own conditional statements and examining them under different circumstances to see how they hold up. This exercise will sharpen your ability to think critically and reason logically.