∀p∀q∀r((p≡T ∧q≡F ∧r≡T) →[(p∧¬q)⨁r≡T])   TRUE        FALSE   because:     The contrapositive:

Q: Which of the following is logically equivalent to: [(￢p ∨ q)∧(￢p ∨q) ∧(￢p ∨r)]→[(p ∨￢q)∧(￢p ∨￢q)…

A: the answers for both mcqs is given below

Q: Prove the following consequent in natural deduction:

A: Proved the given consequent in natural deduction

Q: a)    (p ∧ ~q) ∧ p True False

A: Tell if the statements are true or false. Let p: 3 + 5 = 6; q: 5 + 5 = 10. (a)    (p ∧ ~q) ∧ p True…

Q: 1. Construct a detailed truth table for the following propositional statement:  (((P ∨Q)∧¬P)→Q)…

A: Here in this question we have given two prepositional statement and we have asked to find the…

Q: termine the truth value of each compound statement when p is false, q is false, and r is true.  You…

A: Solution : We have determine the truth value of each compound statement when p is false, q is false,…

Q: Consider the wffs: φ1 ≡ p1 → (p2 → (p3 → p4)) φ2 ≡ (p1 ∧ p2 ∧ p3) → p4 (a) Technically speaking,…

A: Given, φ1 = p1→(p2→(p3→p4)) φ2 = (p1 ^ p2 ^p3)→p4   Rules:  If P is a formula , then ~P is a Well…

Q: What is the truth value for following compound propositions: (Given p=T and q=F) (pv ¬9) → q

A: A truth value is sometimes called a logical value is a value indicating the relation of a…

Q: Find a denial (or the negation) for "(3!x)P(x)".

A: Find a denial (or the negation) for ” (!)P(x)".   We first look at the negation of a statement…

Q: Is the wff valid or invalid? Explain.(∀x)[P(x) ∨ Q(x)]  → (∀x)P(x) ∨ (∀x)Q(i)

A: Here below we use predicate logic rules to check it is valid or not?…

Q: Construct truth tables for the following wffs. Note any tautologies or contradictions. a. A→ (B → A)…

A: Note: The given question contains multiple sub parts solving first three a. b. c.

Q: ollowing formula as much as possible using a series of equivalence laws.                  B ∧ ((B ∧…

A: Here We can simplify the formula using the following equivalence laws: Distributive law: A ∧ (B ∨…

Q: Construct the truth table for each of these compound propositions  (p ↔ q) V (¬p →r) (¬p ↔¬q)↔(q↔p)

A: 1.

Q: Consider the following proof of   ∃y∀xp(x,y)→∀x∃yp(x,y)   1. ¬(∃y∀xp(x,y)→∀x∃yp(x,y))…

A: Lets see the solution in the next steps

Q: Derive the following using natural deduction. (a) PQFPVR QVR (b) P→ (Q → R), P V R, ¬Q → ¬P = R

A: Given,  1. P - ( Q - R )  2. P V R  3. Q - P  . R  4 . P - Q  5. P V R  6. P - ( Q - R )  7 . R…

Q: Give the complete truth table of the following propositional form:   ( ( p ∨ q ) → r ) ↔ ( ( p → q…

A: p q r p∨q (p∨q)→r p→q p→r (p→q)∨(p→r) ((p∨q) → r) ↔ (p→q)∨(p→r) F F F F T T T T T F F T F T T T…

Q: Propositions and quantifiers Let x, y ∈ {2, 3, 4}. For the following propositions provide a proof or…

A: If P stands for a predicate and x denotes an entity, then P(a) indicates the proposition that…

Q: If statement q has the truth value 1, determine all truth value assignments for the primite…

A: Given statement is, (q→[(¬p∨r)∧¬s])∧[¬s→(¬r∧q)] The variables in this statement are p, q, r, s Given…

Q: Determine what rule of replacement was used: (VƆ ~ (Wv X)) ɔ (YvZ) .: ((VƆ ~ (Wv X))·(VƆ ~ (W v…

A: the rule of replacement is given below

Q: Use propositional logic to prove that the following argument is valid. [A → (B → C)] A (A V D') AB →…

A: Given Expression : Given expression : (A -> (B -> C)) ^ (A V D') ^ B -> (D -> C)

Q: Simplify the following assertions (so that ¬ does not appear).

A: Answer to the above question is given below

Q: (¬P ∧ ¬R) ∨ (P ∧ ¬Q ∧ ¬R) is equivalent to ¬R ∧ (Q ⇒ ¬(P ∧ ¬R))

A: TO PROVE LHS = RHS THAT IS    -->  (¬P ∧ ¬R) ∨ (P ∧ ¬Q ∧ ¬R) is equivalent to ¬R ∧ (Q ⇒ ¬(P ∧…

Q: Construct a truth table for each of the following compound propositions. a) p ⊕ (p ∧ q) b) (p ↔ q)…

A: p                         q            p ⊕ (p ∧ q)…

Q: Construct a truth table for each proposition. a.) [(p ∧ q) ∨ r ] ↔ [(p ∧ r ) ∨ (q ∧ r )] b.) [(p ∧…

A: According to the question below the solution:

Q: (а) -(Ал -А) (b) -А — (А > В) (с) AV -A

A: 1) ¬(A^¬A) =¬A V ¬(¬A)              Using DeMorgan's law =¬A V A               = T…

Q: What is (a --> b) and (¬b --> c) and (¬ c↔¬b) logically equivalent to? Use logic laws and label them

A: Solution:   In order to solve this question, first let's understand the neede truth table.

Q: proofs to show the following. P → (Q → R) is provably equivalent to (P ∧ Q) → R ¬(A ∨ B) is provably…

A: It is defined as the simplest form of logic where all the statements are made by propositions. A…

Q: Simplify the following propositions using equivalence laws (B ∨ A) ∨ ¬B ∧ ¬(C ∧ ¬B) ¬(C → B) ∧…

A: According to the information given:- We have to simplify the following propositions using…

Q: 1. Which of the following statements are true? (a) |N|=|Z|  (b) |N||Z| 2. Which of the following…

A: Given the following multiple choice questions we have to answer seperate.

Q: Given the following: (p → q) ∧ ¬r
 (¬p → u) ∨ r
 (¬q → u) → s
 prove s using natural deduction
.

A: modus ponens (p->q) ^ p - > q Resolution (pVq ) ^(~p Vr) -> (qVr) p->q = ~p V q

Q: Th Conjunctive Normal Form (CNF) for (A → B) ∨ (B → A) is: (¬A ∨ B) ∨ (¬B ∨ A)

A: The Conjunctive normal form of  X -> Y => ¬X ∨ Y   Thus using this formula, we need to expand…

Q: Consider the following argument ∀x, N(x) → J(x) ∃x, N(x) ∧ A(x) ∀x, A(x) → E(x) ∴ ∃x, J(x) ∧…

A: We use boolean logic to solve this.

Q: Complete the truth table for the following compound statement. (p→q)↔(q→p)(p→q)↔(q→p) p q…

A: Mathematicians typically employ a binary logic, where each proposition can only take on one of two…

Q: Which  is the logically equivalent to (p ∧ q ∧ r) ∨ (p ∧ ￢q ∧ r) ∨ (￢p ∧ q ∧ r) ∨ (￢p ∧ q ∧ ￢r) ∨…

A: Here the given expression is: (p ∧ q ∧ r) ∨ (p ∧ ￢q ∧ r) ∨ (￢p ∧ q ∧ r) ∨ (￢p ∧ q ∧ ￢r) ∨ (￢p ∧ ￢q ∧…

Q: PROCESSES ENABLES TO DO THINGS FASTER AND IS CONSTRUCTS SYSTEMS IN A MODULAR FORM.

A: EXPLANATION - CONCURRENT PROCESS IN OPERATING SYSTEM - The concurrent processing in the system is…

Q: Conver the following to DNF and CNF form: (A ∨ B) → (C ∧ D)

A: Answer: DNF: ⇒(A∨B)→(C∧D)⇒τA∨B∨(C∧D)⇒A-∧B-∨(C∧D)

Q: Simplify the compound statement ¬(p ∧ (q ∨ r) ∧ ((p ∧ q) → r)).

A: In this question we have been given a compound statement and we need to simplify this statement.

Q: PA¬P = false Pv false = P -(-P) = P P → Q = ¬P V Q

A: There are total 4 expressions given. We have to find the value of P and Q variables. Expressions: P…