Level 3 Logical Reasoning

A group of students is described by the following statements:

1. 1. If Sarah is in the group, then John is also in the group.

2. 2. If Ali is not in the group, then Sarah is not in the group.

Which of the following must be true?

• If Sarah is not in the group, John is not in the group.

• If John is not in the group, then Sarah is not in the group.

• If Ali is in the group, then Sarah and John are in the group.

• If Ali is not in the group, then John is not in the group.

Consider the compound statement:
"If it is raining AND the road is slippery, then the car must slow down."

Which of the following statements is the negation of the compound statement?

• It is not raining OR the road is not slippery OR the car does not slow down.

• It is not raining AND the road is not slippery, and the car slows down.

• If it is raining and the road is slippery, then the car does not slow down.

• The car does not slow down if it is raining and the road is slippery.

Suppose the truth values of p, q, and r are:

• p: True

• q: False

• r: True

What is the truth value of the compound statement:
"(p OR q) AND (NOT q OR r)"?

• True

• False

• none

• True only if r is False

In a debate team, the following conditions hold:

1. If Adam participates, then Ben must participate.

2. If Ben does not participate, then Carol must participate.

Which of the following scenarios is logically impossible?

• Adam does not participate, and Carol participates.

• Adam participates, and Ben does not participate.

• Ben participates, and Carol does not participate.

• Adam does not participate, and Ben does not participate.

If the universal statement "All mathematicians are logical thinkers" is true, which of the following must be true?

• If someone is a logical thinker, they must be a mathematician.

• If someone is not a logical thinker, they cannot be a mathematician.

• If someone is not a mathematician, they cannot be a logical thinker.

• All logical thinkers are mathematicians.

A class of 40 students is described by the following:

1. 30 students like Mathematics.

2. 25 students like Science.

3. 20 students like both Mathematics and Science.

How many students like only Mathematics?

• 10

• 15

• 20

• 25

Let p and q be two statements:

• p: "x is an odd number."

• q: "x is a prime number."

Which of the following is logically equivalent to the statement:
"If x is not odd, then x is not a prime number"?

• If x is a prime number, then x is odd.

• If x is not a prime number, then x is not odd.

• If x is odd, then x is a prime number.

• x is odd AND x is a prime number.

If the statement "If a polygon is a square, then it is a rectangle" is true, which of the following is the contrapositive?

• If a polygon is not a rectangle, then it is not a square.

• If a polygon is a rectangle, then it is a square.

• If a polygon is not a square, then it is not a rectangle.

• If a polygon is not a rectangle, then it is a square.

A teacher observes the following:

1. "If a student studies, then they will pass the exam."

2. "If a student does not pass the exam, then they did not study."

What kind of reasoning connects these two statements?

• Converse reasoning

• Contrapositive reasoning
  

• Inverse reasoning

• Biconditional reasoning

Which of the following best represents the logical equivalence of the compound statement:
"If it is not cold OR it is not raining, then I will go outside"?

• If it is cold AND it is raining, then I will not go outside.

• If I do not go outside, then it is cold OR it is raining.

• If it is not raining AND it is not cold, then I will not go outside.

• If I go outside, then it is cold AND it is raining.