By André Fortin

ISBN-10: 2553009364

ISBN-13: 9782553009365

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Inductive Hypothesis: Assume that the result is true for n = k ( ≥ 1). 4 + 35 . 4 + L k ( k + 2)] + ( k + 1)( k + 3) k ( k + 1)(2k + 7) = + ( k + 1)( k + 3) 6 35 Introduction ( k + 1) [ k (2k + 7) + 6(k + 3)] 6 ( k + 1)(2k 2 + 13k + 18) = 6 ( k + 1)( k + 2)(2k + 9) = 6 = Hence the result follows for all n ∈ Z + , by the principle of Mathematical induction. 47: Prove by induction 1 n ∑ i=1 i(i + 1) = n + 1 n Proof: Assume S ( n): ∑ i=1 n n 1 . = i( i + 1) n + 1 For n = 1, S (1) = 1 1 1 ∑ i=1 i(i + 1) = 1(2) = 1 + 1 1 ⇒ S (1) is true.

54. Assume that the relation R is irreflexive. Is R2 necessarily irreflexive? Give reasons. 55. Given R is a reflexive relation on a set A, show that Rn is reflexive for all positive integers n. 50 Theory of Automata, Formal Languages and Computation 56. Given R is a relation R = {(a, b) a divides b} on the set of positive integers. Determine (a) R–1 (b) R 57. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where ( x, y) ∈ R if and only if (a) xy ≥ 1 (b) x ≡ y (mod b) (c) x = y 2 58.

Degree of vertex: Number of edges having that vertex as an end point. Loop: Graph having an edge from a vertex to itself. Isolated vertex: Vertex with zero as degree. Directed graph: Graph having arrows instead of lines. Outdegree: Number of arrows pointing from a particular node. Indegree: Number of arrows pointing to a particular node. Alphabet: Finite set of symbols. String: Finite sequence of symbols from an alphabet. Lexicographic ordering: Dictionary ordering, except that shorter strings precede longer strings.

### Analyse numerique pour ingenieurs by André Fortin

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