Question

1. Which of the following predicate calculus statements is true?

Question 1 options:

	∀n ∈ ℤ, n + 1 > n
	∃n ∈ ℤ, n + 1 < n
	∀n ∈ ℤ, n > 2n
	∀n ∈ ℤ, 2n > n

2. Which of the following is the correct predicate calculus translation of the sentence "Some natural numbers are at least 100"?

Question 2 options:

	∃n ∈ ℕ, n > 100
	∀n ∈ ℕ, n ≥ 100
	∃n ∈ ℕ, n ≥ 100
	∀n ∈ ℕ, n > 100

3. Which of the following is the correct predicate calculus translation of the sentence "Every rational number is the reciprocal of some other rational number"?

Question 3 options:

	∀p ∈ ℚ, ∃q ∈ ℚ, p = 1/q
	∃p ∈ ℚ, ∀q ∈ ℚ, p = 1/q
	∃p ∈ ℚ, ∃q ∈ ℚ, p = 1/q
	∀p ∈ ℚ, ∀q ∈ ℚ, p = 1/q

4. Which of the following is the negation of the following sentence "Everyone loves chocolate"?

Question 4 options:

	At least one person doesn't love chocolate
	No one loves chocolate
	Someone loves chocolate
	Everyone doesn't love chocolate

5. Assuming F is a set of friends, which of the following is the correct predicate calculus translation of the sentence "Among the group of friends, everyone knows everyone else"?

Question 5 options:

	∀f ∈ F, ∃g ∈ F, f and g know each other
	∀f ∈ F, ∀g ∈ F, f and g know each other
	∃f ∈ F, ∀g ∈ F, f and g know each other
	∃n ∈ ℕ, ∃m ∈ ℕ, f and g know each other

Answer 1

Note: For ease of our reference we consider the first option as (a), the second one as (b), the third one as (c) and the fourth one as (d) for every question.

1.

The correct option is (a).

Option (b) is always false as (n+1) can never be lesser than n.

Option (c) is false in case of non-negative (zero and positive) integers.

Option (d) is false in case of non-positive (zero and negative) integers.

2.

The correct option is (c).

Option (a) is not correct here as we were interested about natural numbers at least (not strictly greater than) 100.

Option (b) is false in case of natural numbers lesser than 100.

Option (d) is false in case of natural numbers lesser than 101.

3.

The correct option is (a) as for each rational number there exists an unique rational number which are reciprocal to each other.

Option (b) is false as one rational number cannot be reciprocal to all rational numbers.

Option (c) is not correct as it is concentrating upon a (not every) rational number (only) and its reciprocal (which is also rational).

Option (d) is not correct as for each rational number all the rational numbers can not be its reciprocal.

4.

The correct option is (a) At least one person doesn't love chocolate.

The logic to get negation of a statement is to find at least a single observation which does not follow the given property. So we need only one person contradicting the fact that "(everyone) loves chocolate".

5.

The correct option is (b).

Option (a) is not correct as it is denoting the event that "in particular, g knows each other corresponding to each f". It is denoting the case that one person (g) knows everyone and is known by everyone. But other that that particular person, a pair of two person may not know each other.

Option (c) is not correct as it is denoting the event that "in particular, f knows each other corresponding to each g". It is again denoting the case that one person (f) knows everyone and is known by everyone. But other that that particular person, a pair of two person may not know each other.

Option (d) is not relevant to the set F at all.