(p^ (~(~pV q))) v (pAq) = p.
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- 6. Show the validity of the logical equivalence:Which of the following is a logical equivalence in first-order logic? A. ¬(∀x)P(x) ≡ (∃x)¬P(x) B. (∀x)(∃y)P(x, y) ≡ (∃y)(∀x)P(x, y) C. (∀x)(P(x) → Q(x)) ≡ (∃x)(P(x) ∧ Q(x)) D. (∀x)(P(x) ∧ Q(x)) ≡ (∀x)(P(x) ∨ Q(x))Prove the following, using the semantic equivalences. In each step apply only one rule once and state the name of the rule. a. (p ∧ q) → r ≡ (p → r ) ∨ (q → r )
- Let the universe of discourse be the set of real numbers. Find the negation of the statement ( for all x,there exists y such that x + y = 0 and xy=1)Which of the following relations complete the statement: If PQ = RS , then 'using the symmetric property of congruence? PQ = QP RS = PQ Option 1 Option 2 РО РО RS = RS Option 3 Option 4ii. Let F(x,y) be the open sentence "x can fool y" where the domain of discourse is the set of all people in the world. Express each of these statements using logical quantifiers. c. Everybody can fool somebody. e. Everyone can be fooled by somebody. f. There is somebody who can be fooled by everybody.
- Using the laws of logical equivalence show:(p → r) ∨ (q → r) ≡ (p ∧ q) → r can you give me the answer becuse the upper answer is for diffrent questionNote: for 2 , you can use ( p ↔ q ≡ (p → q) ∧ (q → p) ) to provef) Proof the following statement using implication proof technique, "If a2 +1 is odd then a a is even".
- (10) Let E denote the set of even integers, and O denote the set of odd integers. (a) Let P = "Every integer is either even or odd." Write P using logic symbols (and common notation for sets, like E, Z, N, Q etc).Using the substitution theorem and the important equivalences (handout) show the following equivalence. Use only one substitution/equivalence rule (such as absorption) per step and justify each step by name. ((-p) → (r V q)) = ((¬r) → ((¬p) → q))(p˄q) → p is a tautology prove using logic laws