Consider the following propositions: • p : You share your solutions with your classmates. • q : You get an academic integrity violation case against you. • r : You passthe course. For each of the following problem, provide the corresponding English sentence. a) p → q b) q → ¬r c) ¬(q ∧ r). d) (p ∧ q) ∨ r e) ¬r ↔ (p ∧ q)

i need answer of all. if any answer will be skipped, your answer will be rejected. only complete answer will be accepted.   b) Make a truth table for the statement ¬P ∧ (Q → P). What can youconclude about P and Q if you know the statement is true? a) Make a truth table for the statement (P ∨ Q) → (P ∧ Q). c) Make a truth table for the statement ¬P → (Q ∧ R).

I need help to answer Question no. 30 prove that the below WFF is a valid argument. (∀x)(∀y)[(P(x) Λ S(x,y)) →Q(y)] Λ (∃x)B(x) Λ (∀x)(B(x) → P(x)) Λ (∀x)(∃y)S(x,y) → (∃x)Q(x) Chapter 1.4 - Predicate Logic Text Book: Discrete Mathematics and its application 7th edition.  Author: Judith L Gersting.

(a) Write the following statement in the symbolic form. Ensure that you explain the stepstaken.If the boss doesn't like me or thinks I am lazy then she will not give me a raise and I willhave to find another apartment.    (b) Determine if the following statements are logically equivalent. Ensure that you explainthe steps taken and the reason for your decision.(i) (¬a → b) ∧ (¬b ∨ (¬a∨ ¬b))(ii) ¬(a ↔ b)   c)Determine if the following statement is a tautology. Ensure that you explain the stepstaken and the reason for your decision.(p ⊕ q ⊕ r) ↔ ((p ∨ q ∨ r) ∧ ¬.(p ∧ r) ∨ (q ∧ r) ∨ (p ∧ q)))

1) Create a truth table and check whether the statement is a tautology or contradiction(?∨~?)↔~(~?∧?)

Exercise 1.4.4: Proving whether two logical expressions are equivalent. About Determine whether the following pairs of expressions are logically equivalent. Prove your answer. If the pair is logically equivalent, then use a truth table to prove your answer. (a) ¬(p ∨ ¬q) and ¬p ∧ q (b) ¬(p ∨ ¬q) and ¬p ∧ ¬q (c) p ∧ (p → q) and p → q (d) p ∧ (p → q) and p ∧ q

Using the predicate symbols shown and the appropriate quantifiers, write each English language statement as a predicate wff. (The domain is the whole world.) B(x): x is a bee. F(x): x is a flower. L(x, y): x loves y All bees love all flowers which one of these answer choices a. (∀x)(B(x) → [(∃y)(F(y) ∧ L(x, y)] b. (∀x)(B(x) → [(∀y)(F(y) ∧ L(x, y)] c. (∀x)(B(x) → [(∀y)(F(y) → L(x, y)] d. (∀x)(B(x) ∧ [(∃y)(F(y) → L(x, y)]

Let p, q, and r be the propositions p: You started a new video game this week. q: You finished all its quests.r: You didn’t win the platinum trophy.Express each of these propositions as an English sentence.¬p → ¬q(p ∧ q) → (¬r)((¬p) → (¬q)) ∨ (q → (¬r))(p ∧ q) ∨ (¬q ∧ r)

A propositional knowledge-base KB consisting of five sentences is given below (note that "/\" is used to denote "logical and", and "\/" is used to denote "logical or", and "~" is used to denote "logical negation"): 1. (~P  /\  ~Q ) -> R2. R -> S3. ~S4. P -> ~U5. U a) Explain in English (in Steps!) that KB |= Q. b) Show resolution steps that leads to KB |- Q. You should first convert the sentences in the KB into clauses.

Consider the following predicates.  P(x) is the statement “x − 1 ≥ 0”. Q(x) is the statement “x + 0 < − 3” R(x) is the statment "x + 1 > 2" Determine the truth value of the following propositions:  (a) ¬P(1) ∧ Q(−2) True or False  (b) Q(−7) ↔ ¬R(6) True or False (c) ¬R(4) → P(−3) True or False