Question

Choose the ordered pair of statements where the first statement implies the second, and the two statements are logically consistent with the main statement.

Main statement: The computer will not work if operating system fails.
Statements: 1. Operating system fails. 2. Operating system does not fail. 3. The computer does not work. 4. The computer works. The ordered pair of statements is:

• A. 2, 1
• B. 2, 3
• C. 1, 4
• D. 4, 2

Hint:

When analyzing conditional statements, remember that the contrapositive is also valid. If "If A, then B" holds, then "If not B, then not A" also holds.

Answer 1

The Correct Option is D

The main statement is "The computer will not work if operating system fails," which can be written as:
\[ \text{If operating system fails, then computer will not work.} \quad (\text{OS fails} \to \text{Computer does not work})
\] This is a conditional statement.
Analyze the options:
- Statement (1) says "Operating system fails", which would imply that the computer does not work. However, it is not immediately relevant to the correct logical pair, as we are looking for a statement indicating that the computer works.
- Statement (4) says "The computer works", which contradicts the main statement that the computer does not work when the operating system fails. This makes the computer working inconsistent with the failure of the operating system. Therefore, if the computer works, the operating system did not fail, i.e., the operating system does not fail.
- Statement (2) says "Operating system does not fail", which directly follows from statement (4) "The computer works" (as the computer will work only if the operating system does not fail). Therefore, statement (4) leads to statement (2).
Thus, the correct pair is (4, 2), where: - "The computer works" (statement 4), - "The operating system does not fail" (statement 2).

Answer 2

The main statement "the computer will not work if operating system fails" fixes one relationship: whenever the operating system fails, the computer is guaranteed not to work. Let's check whether each ordered pair genuinely follows from that rule, taking the first statement as given and asking whether the second must then be true.

1. 2, 1 (Operating system does not fail, so operating system fails): These two statements are opposites of each other. The first can never force the second; this pair is self-contradictory and cannot be right.
2. 2, 3 (Operating system does not fail, so the computer does not work): The main statement only tells us what happens when the operating system fails; it says nothing about what happens when it doesn't fail. So knowing the operating system didn't fail gives no guarantee about whether the computer works or not. This pair assumes more than the main statement actually supports.
3. 1, 4 (Operating system fails, so the computer works): This is the exact opposite of what the main statement says. If the operating system fails, the computer is stated to not work, so concluding that it works contradicts the given rule outright.
4. 4, 2 (The computer works, so operating system does not fail): If the computer is working, it cannot be true that the operating system failed, because the main statement guarantees a failed operating system always stops the computer from working. So a working computer rules out operating system failure, which is exactly what the second statement says. This pair is a forced, valid inference from the main statement.

Only the fourth pair is something that must be true given the main statement, without assuming anything extra.

Therefore, the correct answer is 4, 2.

Answer 3

The main statement "the computer will not work if operating system fails" can be written in symbolic form as \( F \rightarrow \neg W \), where \( F \) stands for "operating system fails" and \( W \) stands for "the computer works." Taking the contrapositive of a true conditional gives an equally true statement: \( W \rightarrow \neg F \), meaning that if the computer works, the operating system did not fail. Let's substitute each ordered pair into these two forms and see which one matches.

1. 2, 1 (\( \neg F \), so \( F \)): This pair asks \( \neg F \) to imply \( F \), a statement implying its own opposite. Neither \( F \rightarrow \neg W \) nor its contrapositive \( W \rightarrow \neg F \) contains any route from \( \neg F \) to \( F \); this pair matches neither form.
2. 2, 3 (\( \neg F \), so \( \neg W \)): Matching this against \( F \rightarrow \neg W \) fails, since the antecedent here is \( \neg F \), not \( F \). Matching it against the contrapositive \( W \rightarrow \neg F \) also fails, since that form starts from \( W \), not \( \neg F \). This pair isn't licensed by either form of the main statement.
3. 1, 4 (\( F \), so \( W \)): The main statement's direct form is \( F \rightarrow \neg W \), meaning \( F \) forces \( \neg W \), the exact opposite of what this pair claims. This pair contradicts the main statement outright.
4. 4, 2 (\( W \), so \( \neg F \)): This matches the contrapositive form \( W \rightarrow \neg F \) exactly, term for term. Since the contrapositive of a true conditional is always true, this pair is a guaranteed, valid inference from the main statement.

Only the fourth pair lines up exactly with the contrapositive of the main statement's symbolic form.

Therefore, the correct answer is 4, 2.