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Chapter 1: Problem 49

"If compound \(X\) is boiling, then its temperature must be a least \(150^{\circ} \mathrm{C}\)." Assuming that this statement is true, which of the following must also be true? a. If the temperature of compound \(X\) is at least \(150^{\circ} \mathrm{C}\), then compound \(X\) is boiling. b. If the temperature of compound \(X\) is less than \(150^{\circ} \mathrm{C}\), then compound \(X\) is not boiling. c. Compound \(X\) will boil only if its temperature is at least \(150^{\circ} \mathrm{C}\). d. If compound \(X\) is not boiling, then its temperature is less than \(150^{\circ} \mathrm{C}\). e. A necessary condition for compound \(X\) to boil is that its temperature be at least \(150^{\circ} \mathrm{C}\). f. A sufficient condition for compound \(X\) to boil is that its temperature be at least \(150^{\circ} \mathrm{C}\).

Short Answer

Expert verified
Based on the given statement, the following must also be true: b. If the temperature of compound $X$ is less than $150^{\circ} \mathrm{C}$, then compound $X$ is not boiling. c. Compound $X$ will boil only if its temperature is at least $150^{\circ} \mathrm{C}$. e. A necessary condition for compound $X$ to boil is that its temperature be at least $150^{\circ} \mathrm{C}$.

Step by step solution

01

Rephrase the Given Statement

We have a true statement: "If compound X is boiling, then its temperature must be at least 150掳C." This can be rephrased as: "Boiling of compound X implies that its temperature is at least 150掳C."
02

Analyze the Options

Now, let's break down each option and compare it with the true statement. a. "If the temperature of compound X is at least 150掳C, then compound X is boiling." This statement is the converse of the given true statement, and its truth value is not necessarily the same. b. "If the temperature of compound X is less than 150掳C, then compound X is not boiling." This statement is the contrapositive of the given true statement, and logically, it has the same truth value as the original statement. c. "Compound X will boil only if its temperature is at least 150掳C." This statement is equivalent to the given true statement. d. "If compound X is not boiling, then its temperature is less than 150掳C." This statement is the inverse of the given true statement, and its truth value is not necessarily the same. e. "A necessary condition for compound X to boil is that its temperature be at least 150掳C." Being a necessary condition means it must happen for the event to occur. The given true statement says that if compound X is boiling, then the temperature is at least 150掳C, so a temperature of at least 150掳C is necessary for compound X to boil. f. "A sufficient condition for compound X to boil is that its temperature be at least 150掳C." Being a sufficient condition means it is enough for the event to occur, but not necessarily required. From the given true statement, we only know that boiling implies a temperature of at least 150掳C. We cannot say that having a temperature of at least 150掳C is sufficient for compound X to boil.
03

Conclusion

Based on our analysis, options b, c, and e are true statements that must also be true if the given statement is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Statements
Imagine you have a simple rule: if an event A happens, then event B follows. This is the essence of a conditional statement. In logic, we express this as 'If A, then B', symbolically written as 'A 鈫� B'. This statement implies that whenever A occurs, B must also occur. However, the reverse is not implied - B can occur without A.

In the context of the provided exercise, the conditional statement is: 'If compound X is boiling, then its temperature must be at least 150掳C.' It's essential to note that while this rule holds, the reverse isn't stated; a temperature of 150掳C doesn't necessarily mean that compound X will boil, as other factors might be involved.
Converse
The converse of a conditional statement flips the hypothesis and the conclusion. For the statement 'If A, then B' (A 鈫� B), the converse is 'If B, then A' (B 鈫� A). Importantly, the truth of the original statement does not guarantee the truth of its converse.

For instance, in the exercise, the converse of the given statement is option a: 'If the temperature of compound X is at least 150掳C, then compound X is boiling.' We cannot confirm its truth without additional information, indicating that the converse does not necessarily share the truth value of the original statement.
Contrapositive
The contrapositive also flips the hypothesis and conclusion of a conditional statement but with a twist: it also negates both. So, the contrapositive of 'If A, then B' (A 鈫� B) is 'If not B, then not A' (卢B 鈫� 卢A). A key point is that the contrapositive always has the same truth value as the original statement.

In our exercise, option b is the contrapositive: 'If the temperature of compound X is less than 150掳C, then compound X is not boiling,' which must also be true if our initial statement is true.
Inverse
The inverse of a conditional statement negates both the hypothesis and the conclusion of the original. So, for 'If A, then B' (A 鈫� B), the inverse is 'If not A, then not B' (卢A 鈫� 卢B). Like the converse, the truth of the inverse is not guaranteed by the truth of the original statement.

In the case of the exercise, option d presents the inverse: 'If compound X is not boiling, then its temperature is less than 150掳C.' This cannot be counted as true solely based on the original statement, illustrating the separate nature of the inverse's truth.
Necessary and Sufficient Conditions
A necessary condition is something that must be true for a statement to hold but on its own may not be enough to guarantee the statement. In our example, having a temperature of at least 150掳C is necessary for compound X to boil, as mentioned in option e.

On the other hand, a sufficient condition ensures that a statement holds, but it may not be the only way that the statement can be true. To say that a condition is sufficient, we must be sure that it always leads to the outcome. Option f suggests that a temperature of at least 150掳C is sufficient for compound X to boil, but based on the information given, we cannot confirm this as being sufficient鈥攊t's possible that more is needed for boiling to occur.

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Most popular questions from this chapter

Use modus ponens or modus tollens to fill in the blanks in the arguments of 1-5 so as to produce valid inferences. If this is a while loop, then the body of the loop may never be executed. ______________ The body of the loop may never be executed.

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If statement forms \(P\) and \(Q\) are logically equivalent, then \(P \leftrightarrow Q\) is a tautology. Conversely, if \(P \leftrightarrow Q\) is a tautology, then \(P\) and \(Q\) are logically equivalent. Use \(\leftrightarrow\) to convert each of the logical equivalences in 29-31 to a tautology. Then use a truth table to verify each tautology. $$ p \rightarrow(q \rightarrow r) \equiv(p \wedge q) \rightarrow r $$

In logic and in standard English, a double negative is equivalent to a positive. Is there any English usage in which a double positive is equivalent to a negative? Explain.

"Do you mean that you think you can find out the answer to it?" said the March Hare. "Exactly so," said Alice. "Then you should say what you mean," the March Hare went on. "I do," Alice hastily replied; "at least-at least I mean what I say - that's the same thing, you know." "Not the same thing a bit"" said the Hatter. "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!" \- from "A Mad Tea-Party" in Alice in Wonderland, by Lewis Carroll The Hatter is right. "I say what I mean" is not the same thing as "I mean what I say." Rewrite each of these two sentences in if-then form and explain the logical relation between them. (This exercise is referred to in the introduction to Chapter 3.)
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