Abstract Algebra Theory and Applications - Computer Science ...

1.1 MATHEMATICAL INDUCTION 25We have an equivalent statement of the Principle of Mathematical Inductionthat is often very useful:Second Principle of Mathematical Induction. Let S(n) be a statementabout integers for n ∈ N and suppose S(n 0 ) is true for some integer n 0 . IfS(n 0 ), S(n 0 +1), . . . , S(k) imply that S(k+1) for k ≥ n 0 , then the statementS(n) is true for all integers n greater than n 0 .A nonempty subset S of Z is well-ordered if S contains a least element.Notice that the set Z is not well-ordered since it does not contain a smallestelement. However, the natural numbers are well-ordered.Principle of Well-Ordering. Every nonempty subset of the natural numbersis well-ordered.The Principle of Well-Ordering is equivalent to the Principle of MathematicalInduction.Lemma 1.1 The Principle of Mathematical Induction implies that 1 is theleast positive natural number.Proof. Let S = {n ∈ N : n ≥ 1}. Then 1 ∈ S. Now assume that n ∈ S;that is, n ≥ 1. Since n+1 ≥ 1, n+1 ∈ S; hence, by induction, every naturalnumber is greater than or equal to 1.□Theorem 1.2 The Principle of Mathematical Induction implies that thenatural numbers are well-ordered.Proof. We must show that if S is a nonempty subset of the natural numbers,then S contains a smallest element. If S contains 1, then the theoremis true by Lemma 1.1. Assume that if S contains an integer k such that1 ≤ k ≤ n, then S contains a smallest element. We will show that if a set Scontains an integer less than or equal to n+1, then S has a smallest element.If S does not contain an integer less than n + 1, then n + 1 is the smallestinteger in S. Otherwise, since S is nonempty, S must contain an integer lessthan or equal to n. In this case, by induction, S contains a smallest integer.□Induction can also be very useful in formulating definitions. For instance,there are two ways to define n!, the factorial of a positive integer n.• The explicit definition: n! = 1 · 2 · 3 · · · (n − 1) · n.• The inductive or recursive definition: 1! = 1 and n! = n(n − 1)! forn > 1.