SAT Logic

One of the most important ideas in logic is that of implication. We say \( A \) implies \( B \), written as \( A \implies B \), if the truth of \(A\) makes the truth of \( B \) necessary. For example, the statement "If I can see the sun in the sky, it must be day," is a statement of implication. "Seeing the sun in the sky" implies "it is day." Statements of implication are frequently written in "if, then" form.

On the SAT, logic problems present you with several statements and ask about the logical conclusions that can be drawn from them. Understand what the terms Converse and Contrapositive mean, and you will be one step closer to getting logic problems right.

Converse: The converse switches the position of the terms:

Statement: \( P \implies Q \)

Converse: \( Q \implies P \)

Contrapositive: The contrapositive negates both terms in an implication and switches their positions. A statement is logically equivalent to its contrapositive:

Statement: \( P \implies Q \)

Contrapositive: NOT \( Q \implies\) NOT \( P\)

If Sally goes to the store, then Edward will go to the lake. If Edward goes to the lake, then Freddy will go home. If Sally goes to the store, which of the following statements must be true?

\(\begin{array}{r r l} & \text{I.} & \text{Edward goes to the lake. }\\
& \text{II.} & \text{Freddy goes home.}\\
& \text{III.} & \text{Freddy does not go home.}\\
\end{array}\)

(A)\(\ \ \) I only
(B)\(\ \ \) II only
(C)\(\ \ \) III only
(D)\(\ \ \) I and II only
(E)\(\ \ \) I and III only

Correct Answer: D

Solution:

Since Sally goes to the store, we know Edward will go to the lake. Thus option I is true. Since Edward goes to the lake, we know that Freddy will go home. Thus option II is true. Since option II is true, we know that option III is false.

Thus, only statements I and II must be true.

Incorrect Choices:

(A), (B), (C), and (E)
The solution explains why these choices are wrong.

If Alex goes dancing, then Betty goes dancing.
If Betty goes dancing, then Charlie goes dancing.
If Charlie goes dancing, then Daniel does not go dancing.
If Daniel goes dancing, then Ellen does not go dancing.

If we know that Alex goes dancing, which of the following statements must be true?

(A)\(\ \ \) Betty does not go dancing.
(B)\(\ \ \) Charlie does not go dancing.
(C)\(\ \ \) Daniel does not go dancing.
(D)\(\ \ \) Ellen does not go dancing.
(E)\(\ \ \) Ellen goes dancing.

Correct Answer: C

Solution:

If Alex goes dancing, then Betty goes dancing, so (A) is false.
Since Betty goes dancing, Charlie goes dancing, so (B) is false.
Since Charlie goes dancing, Daniel does not go dancing, so (C) is true.
Since Daniel does not go dancing, we do not know if Ellen will be dancing or not, so neither (D) nor (E) must be true.

Hence, the correct answer is (C).

Incorrect Choices:

(A), (B), (D), and (E)
The solution explains why these choices are wrong.

If I do not study hard, I will not pass the course.
If I do not pass the course, I cannot graduate this year.
If I cannot graduate this year, I will have to repeat the academic year.

Which of the following statements must be true?

(A) If I study hard, I will graduate this year.
(B) If I graduated this year, then I studied hard.
(C) If I passed the course, then I will graduate this year.
(D) If I have to repeat the academic year, then I did not pass the course.
(E) If I have to repeat the academic year, then I passed the course.

Correct Answer: B

Solution:

Let's analyze the statements one at a time.

(A) It is possible that I studied hard but still did not pass the course, so I cannot graduate this year. (A) is false.
(B) Given that I graduated this year, it means that I passed the course and hence that I studied hard. Thus (B) is true.
(C) Even if I passed the course, that does not guarantee that I can graduate this year. Thus (C) need not be true.
(D) / (E) If I have to repeat the academic year, then that doesn't tell us if I could graduate this year, and hence it doesn't tell us if I passed the course. Thus (D) and (E) need not be true.

Incorrect Choices:

(A), (C), (D), and (E)
The solution explains how to eliminate these choices.