Let K(x, y) be the statement "x knows y", where the universe of discourse for both x and y is the set of all people. (Assume it is possible for x to know y while y does not know x, and hence K(x, y) and K(y,x) are different statements.) Translate the sentence "there are two distinct persons who know each other, and no one else" into First Order Logic.

1. Explain the difference between first-order logic and propositional logic.  2.  The language of first-order logic is based on objects and relations. An example of a first-order logic expression is  "squares near the WAMPUS are smelly." What are the objects and relations in that expression?  3. What is an atomic sentence? Give an example.  4. What is the universal quantifier? Write the sentence "All kings are persons" using the universal quantifier and other logic symbols.      Hint: To find the logic symbols in blackboard click on fx from the toolbar above and select the symbols tab. 5. What is the existential quantifier? Write the sentence "There is a prime number that is even" using the existential quantifier and other logic symbols.

Let the domain be the set of all animals. Define the following predicates: A(x) : x can add. • W(x) : x can waltz. • D(x) : x is a dog. • C(x) : x is a cat. Translate each of the following logical expressions into English. 1. Vx(D(x) → (A(x) V W(x)). 2. Ex(C(x) A ¬A(x) ^ W(x)) 3. -3x(D(x) ^ A(x)) 4. Vx((D(x) V C(x)). → W(x))

PLEASE HELP!!!

3. Express the following English sentences using propositional functions, quantifiers and logical connections. (a) Someone in the class is helped by everyone in the class. (b) There is a hard working student in the class who has not been helped by any student in class. (c) For everyone in the class, someone who is smart has helped him/her. SO (d) There are two people in the class who have not helped each other. (e) There is a hard working student in the class who is helped by every smart student in class. (f) There are some companies that hire only hard working students.

Set up truth tables for the statementsP ⇒ (P ⇒ Q), (P ⇒ Q) ⇒ Q, P ∧ (∼ Q), (∼ P) ∨ (∼ Q)Are any of these statements equivalent?

Formalize the following sentence in english: 3X. elephant(X)|

Prove the following predicate logic statements valid or invalid: A) ∀x(A(x) → B(x)) ∧ ∀x(A(x) ∨ ¬C(x)) ∧ ∃x(¬B(x)) → ¬∃x(¬A(x) ∧ ¬B(x) ∧ ¬C(x)) B) ∀x(A(x) → B(x)) ∧ ∃x(A(x) ∨ B(x)) → ∃x(A(x) ∧ B(x))

Using predicate logic, prove that the following argument is valid. Provide the translation and the proof sequence. Every computer science student works harder than somebody, and everyone who works harder than any other person gets less sleep than that person.  Riley is a computer science student.  Therefore, Riley gets less sleep than someone else. Use the symbols, C(x), W(x,y), S(x,y), r.

Translate each of these nested quantifications into an English statement that expresses a mathematicalfact. The domain in each case consists of all real numbers.a) ∃x∀y(x + y = y) b) ∀x∀y (((x ≥ 0) ∧ (y < 0)) → (x − y > 0))c) ∃x∃y(((x ≤ 0) ∧ (y ≤ 0)) ∧ (x − y > 0))d) ∀x∀y((x ≠ 0) ∧ (y ≠ 0) ↔ (xy ≠ 0))