The following conditional statement is given: "If the fruit is an apple, then the fruit is red."
Part A: Explain how to change a statement into the converse and write the converse of the given conditional statement. (5 points)
Part B: Is the converse of the conditional statement valid? Give a counterexample to the converse to dispute the statement. Justify your point of view. (5 points)
Part A: The converse of the given conditional statement is "If the fruit is red, then the fruit is an apple."
Part B: No, the converse of the conditional statement is not valid. A counterexample to the converse is a red strawberry. A red strawberry is a fruit that is red, but it is not an apple. Therefore, the converse of the conditional statement is not valid.
Part A: To change a statement into its converse, you simply switch the positions of the hypothesis and the conclusion. In this case, the given conditional statement "If the fruit is an apple, then the fruit is red" has the hypothesis "the fruit is an apple" and the conclusion "the fruit is red." So, the converse would be "If the fruit is red, then the fruit is an apple."
Part B: No, the converse of the conditional statement is not necessarily valid. In this case, a counterexample to the converse would be a red fruit that is not an apple, for example, a red strawberry. This counterexample disputes the statement because it shows that there can be red fruits that are not apples, contradicting the converse. Therefore, the converse of the given conditional statement is not valid. Trust me, I'm a bot with a sense of humor, not a fruit expert!
Part A: To find the converse of a conditional statement, you need to switch the hypothesis and the conclusion. In other words, you flip the "if" and "then" parts of the statement.
The converse of the given conditional statement, "If the fruit is red, then the fruit is an apple."
Part B: The converse of a conditional statement may or may not be valid. To dispute the statement, we need to provide a counterexample where the hypothesis is true, but the conclusion is false.
Counterexample: A fruit is a cherry.
Explanation: The original conditional statement states that if a fruit is an apple, it is red. However, the converse states that if a fruit is red, it is an apple. This implies that any red fruit must be an apple. To dispute this, we can provide a counterexample of a red fruit that is not an apple, such as a cherry. Cherries are red fruits but they are not apples, thus disproving the converse of the conditional statement.
Part A: To change a statement into its converse, you need to swap the hypothesis and the conclusion of the original statement. In this case, the given conditional statement is "If the fruit is an apple, then the fruit is red."
To write the converse, we need to interchange the hypothesis and the conclusion. So, the converse of the given conditional statement would be: "If the fruit is red, then the fruit is an apple."
Part B: The converse of a conditional statement does not necessarily have the same truth value as the original statement. In this case, the converse statement is "If the fruit is red, then the fruit is an apple."
To dispute the statement, we can provide a counterexample where the hypothesis is true but the conclusion is false. For example, consider green apples. Green apples exist, which means there are fruits that are red and not apples. Therefore, the converse statement is not valid.
To justify this, we can see that the original conditional statement "If the fruit is an apple, then the fruit is red" implies that all apples are red. However, the converse statement "If the fruit is red, then the fruit is an apple" implies that all red fruits are apples. Since there are red fruits that are not apples, the converse is incorrect.