Algorithmic foundations.docx
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This document summarizes three key points: 1) It proves the logical equivalences of three expressions: (a) ¬ (p∨ (q∨¬r)) ∧q≡ ((¬p ∧ q) ∧ R), (b) (p→r) ∨ (q→r) ≡ (p∧q) →r, and (c) (∀×∈⋃ (P(x) →¬Q(x)) ≡ ¬∃×∈⋃. (P(x) ∧Q(x)). 2) It analyzes several expressions involving predicates like D(x
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- 1. Algorithmic Foundations 1 .Usingthe laws of logical equivalence show: (a) ¬ (p∨ (q∨¬r)) ∧q≡ ((¬p ∧ q) ∧ R) solution………………………………………………………………………………………………………………… ……………………………………………………………… Hence the proof that ¬ (p∨ (q∨¬r)) ∧q is equivalent to ((¬p ∧ q) ∧ R) (b)(p→r)∨ (q→r) ≡ (p∧q) →r p q r ((p → r) ∨ (q → r)) p q r ((p ∧ q) → r) 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 Call +25479 Call +254791410785 for the Call +254791410785 for the solution for the solution 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 Hence the proof that (p→r) ∨ (q→r) ∧q is equivalent to (p∧q) →r (c) (∀×∈⋃ (P(x) →¬Q(x)) ≡ ¬∃×∈⋃. (P(x) ∧Q(x)) P(x) P(x) 0 0 Call/WhatsApp +254791410785 1
- 2. P(x) = Q(x) = (P(x) ∧Q(x) =…………………………………………………………………………………………….... Now looking at the truth table P(x) is equivalent Q(x) and (∀×∈⋃ (P(x) →¬Q(x)) is equivalent to ¬∃×∈⋃. (P(x) ∧Q(x)) hence the proof. 2. Assuming thefollowing predicates: •L(x, y):x is strictly less than y (x<y); •E(x):x is even; •P(x): x is a prime number; •EQ(x, y):x equals y (x=y); •G(x):x isgreater than zero (x>O); •D(x, y): x divides y exactly; Determine which of the followingformulae are true (in your answer include an expression of the formula inconcise (good) English without variables). (a) ∀×∈N. (D(2, x) →E(x)) Given that x/2= is an integer____________________hence x is even hence the proof. (b) ∃y ∈N. ∀×∈N.L(x, y) (c) ∃×∈Z+.∀y ∈Z+.(G(x) →(P(y) ∧L(x, y))) Next,usingtheabove predicates and quantifiers were necessary, express the followingEnglish statements inlogic. (d) "Any non-zero integer divides itself" (e) "Aprime number's only positive factors are 1and itself." 3. Provethat (BA) ⋃ (CA) =(B ⋃C)A using: (a) A containment proof. (b) Using set builder notation and logical equivalences. 4. For each of the following functions find the inverse or explain why no inverse exists. a. f :N→ Nwhere f(x) =4·x2 + 1 b. g : Z→ Z where g(x) = x + 7 Call/WhatsApp +254791410785 for customized the for customized the1