Discrete Math with Proof (1st Edition) by Eric Gossett

By Eric Gossett

This booklet equips readers to use discrete arithmetic and offers possibilities for perform of the techniques provided. insurance of algorithms is integrated. Combinatorics gets extra assurance than in different books.

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193. Prove that if f (n) = Ω1 (g(n)), then f (n) = Ω2 (g(n)), or find a counterexample to this claim. 194. Prove that if f (n) = Ω2 (g(n)), then f (n) = Ω1 (g(n)), or find a counterexample to this claim. 195. Prove or disprove: If f (n) = O(g(n)), then f (n) = Ω(g(n)). If f (n) = O(g(n)), then f (n) = Ω2 (g(n)). 196. Define the relation ≡ by f (n) ≡ g(n) iff f (n) = Ω(g(n)) and g(n) = Ω(f (n)). Similarly, define the relation ≡2 by f (n) ≡ g(n) iff f (n) = Ω2 (g(n)) and g(n) = Ω2 (f (n)). Show that ≡ is an equivalence relation, but ≡2 is not an equivalence relation.

X := x + y y := x − y x := x − y function increment(y) comment Return y + 1, where y ∈ IN x := 0; c := 1; d := 1; while (y > 0) ∨ (c > 0) do a := y mod 2; if a ⊕ c then x := x + d; c := a ∧ c; d := 2d; y := y/2 ; return(x) Prove that the following algorithm for the multiplication of natural numbers is correct. 48 Chap. 5. Correctness Proofs 1. 2. 3. 4. 5. 253. Prove that the following algorithm for the multiplication of natural numbers is correct. 1. 2. 3. 4. 5. 254. function multiply(y, z) comment Return yz, where y, z ∈ IN x := 0; while z > 0 do x := x + y · (z mod 2); y := 2y; z := z/2 ; return(x) Prove that the following algorithm for the multiplication of natural numbers is correct, for all integer constants c ≥ 2.

T (1) = 1, and for all n ≥ 2, T (n) = 2T (n − 1) + n2 − 2n + 1. 230. T (1) = 1, and for all n ≥ 2, T (n) = n · T (n − 1) + n. 39 Sec. 3. 1]). The following are interesting variants of this recurrence. 231. State and prove a general formula for recurrences of the form T (n) = 232. State and prove a general formula for recurrences of the form T (n) = 233. d aT (n/c) + bn2 if n ≤ 1 otherwise. State and prove a general formula for recurrences of the form T (n) = 234. d if n ≤ 1 aT (n/c) + b otherwise.

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