State division and remainder algorithm. Suppose that the domain of the propositional function P(x) consists of the integer 0, 1, 2, 3 and 4. Write out each of the following propositions using disjunctions, conjunctions and negations. ∃x P(x) ∀x P(x) ∃x ¬P(x) ∀x ¬P(x) ¬∃x P(x) ¬∀x P(x)

Question

State division and remainder algorithm. Suppose that the domain of the propositional function P(x) consists of the integer 0, 1, 2, 3 and 4. Write out each of the following propositions using disjunctions, conjunctions and negations.  ∃x P(x) ∀x P(x) ∃x ¬P(x) ∀x ¬P(x) ¬∃x P(x) ¬∀x P(x)

Suresh Chand · Accepted Answer

Division and Remainder Algorithm: Let a be an integer and 'd' be positive integer then there exists two unique integer q and r with 0 < r < d such that a = dq + r where a = dividend d = divisior q = quotient r = remainder i.e. dividend = divisor * quotient + Remainder For Example: let a = 19, d = 3 3 ) 19 ( 6 -18 ----- 1 19 = 3 x 6 + 1 Second Part: As we know that Universal Quantifier can be expressed as conjunctions and Existential Quantifier can be expressed as Disjunctions, given functions can be written as following: a) ∃x P(x) = P(0) ∨ P(1) ∨ P(2) ∨ P(3) ∨ P(4) b) ∀x P(x) = P(0) ∧ P(1) ∧ P(2) ∧ P(3) ∧ P(4) c) ∃x ¬P(x) =  ¬P(0) ∨ ¬P(1) ∨ ¬P(2) ∨ ¬P(3) ∨ ¬P(4) d) ∀x ¬P(x) = ¬P(0) ∧ ¬P(1) ∧ ¬P(2) ∧ ¬P(3) ∧ ¬P(4) e) ¬∃x P(x) = ¬ (P(0) ∨ P(1) ∨ P(2) ∨ P(3) ∨ P(4)) f) ¬∀x P(x) = ¬ (P(0) ∧ P(1) ∧ P(2) ∧ P(3) ∧ P(4))

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