Principle of Mathematical Induction - Analysis | MATH 554, Study notes of Mathematics

Download Principle of Mathematical Induction - Analysis | MATH 554 and more Study notes Mathematics in PDF only on Docsity! PRINCIPLE OF MATHEMATICAL INDUCTION (MI) Let N = {1, 2, 3, . . . } be the natural numbers. Let Z = {. . . ,−2,−1, 0, 1, 2, . . . } be the integers. Let P (n) be a statement (that is either true or false) about n. Theorem 1.2.1: PMI (basic form) If base step: P (1) is true inductive step: for each n ∈ N: [ P (n) is true ] =⇒ [ P (n + 1) is true ] then P (n) is true for each n ∈ N. The proof of the PMI is based Peano’s Postulates of N. Theorem 1.2.2: PMI (doesn’t matter where you start form) Fix n0 ∈ Z. If base step: P (n0) is true inductive step: for each n ∈ Z with n ≥ n0: [ P (n) is true ] =⇒ [ P (n + 1) is true ] then P (n) is true for each n ∈ Z such that n ≥ n0. Theorem 1.2.3: PMI (strong form) Fix n0 ∈ Z. If base step: P (n0) is true inductive step: for each n ∈ Z with n ≥ n0: [ P (j) is true for j = n0, 1 + n0, . . . , n ] =⇒ [ P (n + 1) is true ] then P (n) is true for each n ∈ Z such that n ≥ n0. 1