# 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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Example text

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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