﻿ Exercises - Logic - MISCELLANEOUS TOPICS - SAT SUBJECT TEST MATH LEVEL 1 ﻿

## CHAPTER 19Logic

### Exercises

1. If the statement “Every perfect set is dense” is false, which of the following statements must be true?

(A) Some perfect sets are dense.

(B) No perfect sets are dense.

(C) All perfect sets are not dense.

(D) Some perfect sets are not dense.

(E) No dense sets are perfect.

2. If the statement “Some sloths have three toes” is true, which of the following statements must be false?

(A) All sloths have three toes.

(B) No sloths have three toes.

(C) Some sloths do not have three toes.

(D) All three-toed animals are sloths.

(E) Some three-toed animals are sloths.

3. Which of the following statements is equivalent to the statement “If it is sunny, I will go to the beach”?

(A) If I go to the beach, it is sunny.

(B) If I don’t go to the beach, it isn’t sunny.

(C) If it isn’t sunny, I won’t go to the beach.

(D) It is sunny, and I go to the beach.

(E) It isn’t sunny, and I don’t go to the beach.

4. Which of the following statements is equivalent to “If 0 < x < 1, then  is irrational”?

(A) If 0 < x < 1, then  is rational.

(B) If  is irrational, then 0 < x < 1.

(C) If  is rational, then 0 < x < 1.

(D) If  is rational, then x  0 or x  1.

(E) If x  0 or x  1, then  is rational.

5. John said, “Nicole and Caroline each passed today’s test.” If John’s statement is false, which of the following statements must be true?

(A) If Nicole passed, then Caroline passed.

(B) If Nicole passed, then Caroline failed.

(C) Nicole passed, but Caroline failed.

(D) Nicole failed and Caroline failed.

(E) Either Nicole or Caroline failed.

6. Which of the following is equivalent to the statement, “If John is eligible to vote, then John is a citizen”?

(A) All citizens are eligible to vote.

(B) If John is not eligible to vote, then John is not a citizen.

(C) Some citizens are not eligible to vote.

(D) If John is not a citizen, then John is not eligible to vote.

(E) If John is a citizen, then John is eligible to vote.

 1. (D) 3. (B) 5. (E) 2. (B) 4. (D) 6. (D)

Solutions

Each of the problems in this set of exercises is typical of a question you could see on a Math 1 test. When you take the model tests in this book and, in particular, when you take the actual Math 1 test, if you get stuck on questions such as these, you do not have to leave them out—you can almost always answer them by using one or more of the strategies discussed in the “Tactics” chapter. The solutions given here do not depend on those strategies; they are the correct mathematical ones.

See Important Tactics for an explanation of the symbol ⇒, which is used in several answer explanations.

1. (D) To answer the question, you do not need to know what is a perfect set or what is a dense set. You must simply use KEY FACT R2: if it is not true that all perfect sets are dense, then some (at least one) perfect sets are not dense.

2. (B) By KEY FACT R3, the negation of the statement “Some sloths have three toes” is “No sloth has three toes” (choice B). Since the negation of a true statement is false, B must be false. Note that E is true and that from the given statement, it is impossible to know whether A, C, and D are true or false.

3. (B) By KEY FACT R4, a conditional statement (p  q) is equivalent to its contrapositive (q  p). Therefore, “Sunny  Beach” is equivalent to “no Beach  not Sunny.”

4. (D) Note that the original statement is false

In fact, all five choices are false, but the only one that is equivalent to the original is its contrapositive.

If p is the statement “0 < x < 1” and q is the statement “  is irrational,” then p is “x  0 or x  1” and q is “  is rational.” So the contrapositive (q  p) is choice D:

If  is rational, then x  0 or x  1.

5. (E) If you want to disprove the statement “Nicole passed and Caroline passed,” you have to show that either Nicole failed or Caroline failed. Of course, John’s statement would be false if they each failed, but John’s statement is false even if only one of them failed.

6. (D) Any conditional statement (p  q) is equivalent to its contrapositive (q  p). The contrapositive of “If John is eligible to vote, then John is a citizen” is “If John is not a citizen, then John is not eligible to vote.” None of the othet choices is equivalent to the given statement.

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