Problem 1. [24 points]
Translate the following sentences from English to predicate logic. The domain that you are
working over is X, the set of people. You may use the functions S(x), meaning that “x has
been a student of 6.042,” A(x), meaning that “x has gotten an ‘A’ in 6.042,” T(x), meaning
that “x is a TA of 6.042,” and E(x, y), meaning that “x and y are the same person.”
(a) [6 pts] There are people who have taken 6.042 and have gotten A’s in 6.042
(b) [6 pts] All people who are 6.042 TA’s and have taken 6.042 got A’s in 6.042
(c) [6 pts] There are no people who are 6.042 TA’s who did not get A’s in 6.042.
(d) [6 pts] There are at least three people who are TA’s in 6.042 and have not taken 6.042
(a) ∃x | A(x)
any ideas on the others?
On my side i think the answer is the same as yours
(a) ∃x (S(x) ∧ A(x))
(b) ∀x (T(x) ∧ S(x) → A(x))
(c) ∀x (T(x) → A(x))
(d) ∃x ∃y ∃z (T(x) ∧ T(y) ∧ T(z) ∧ ¬S(x) ∧ ¬S(y) ∧ ¬S(z))
To translate the sentences from English to predicate logic, we can use the given functions and symbols. Let's break down each part of the sentences and convert them into logical statements.
(a) There are people who have taken 6.042 and have gotten A's in 6.042.
To translate this into predicate logic, we can use the following notation and symbols:
- Domain: X (set of people)
- Functions:
- S(x): x has been a student of 6.042
- A(x): x has gotten an 'A' in 6.042
The logical statement can be written as:
∃x (S(x) ∧ A(x))
Explanation: Here, ∃x is the existential quantifier, which means there exists an x such that the condition holds. S(x) represents that x has been a student of 6.042, and A(x) represents that x has gotten an 'A' in 6.042.
(b) All people who are 6.042 TA's and have taken 6.042 got A's in 6.042.
To translate this into predicate logic, we can use the following symbols:
- Domain: X (set of people)
- Functions:
- T(x): x is a TA of 6.042
- S(x): x has been a student of 6.042
- A(x): x has gotten an 'A' in 6.042
The logical statement can be written as:
∀x ((T(x) ∧ S(x)) → A(x))
Explanation: Here, ∀x is the universal quantifier, which means for all x, the condition holds. T(x) represents that x is a TA of 6.042, S(x) represents that x has been a student of 6.042, and A(x) represents that x has gotten an 'A' in 6.042. The arrow symbol (→) represents implication, meaning that if someone is a TA and has taken 6.042, they got an 'A' in 6.042.
(c) There are no people who are 6.042 TA's who did not get A's in 6.042.
To translate this into predicate logic, we can use the following symbols:
- Domain: X (set of people)
- Functions:
- T(x): x is a TA of 6.042
- A(x): x has gotten an 'A' in 6.042
The logical statement can be written as:
¬∃x (T(x) ∧ ¬A(x))
Explanation: Here, ¬∃x represents negation of the statement - there does not exist an x such that the condition holds. T(x) represents that x is a TA of 6.042, and ¬A(x) represents that x did not get an 'A' in 6.042.
(d) There are at least three people who are TA's in 6.042 and have not taken 6.042.
To translate this into predicate logic, we can use the following symbols:
- Domain: X (set of people)
- Functions:
- T(x): x is a TA of 6.042
- S(x): x has been a student of 6.042
The logical statement can be written as:
∃x ∃y ∃z (T(x) ∧ T(y) ∧ T(z) ∧ (x ≠ y) ∧ (x ≠ z) ∧ (y ≠ z) ∧ (¬S(x) ∧ ¬S(y) ∧ ¬S(z)))
Explanation: Here, ∃x ∃y ∃z represents the existential quantifier for three different variables x, y, and z, meaning that there exist at least three x, y, and z such that the condition holds. T(x) represents that x is a TA of 6.042, ¬S(x) represents that x has not taken 6.042. The conditions (x ≠ y), (x ≠ z), and (y ≠ z) ensure that the three people are different from each other.