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- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/10%3A_Some_Prerequisite_Topics/10.02%3A_Well_Ordering_and_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/04%3A_New_Page/4.01%3A_New_PageIn this chapter we discuss the principle of mathematical induction. Be aware that the word induction has a different meaning in mathematics than in the rest of science. The principle of mathematical i...In this chapter we discuss the principle of mathematical induction. Be aware that the word induction has a different meaning in mathematics than in the rest of science. The principle of mathematical induction depends on the order structure of the natural numbers, and gives us a powerful technique for proving universal mathematical claims.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/09%3A_Appendices/9.04%3A_Well_Ordering_and_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/10%3A_Some_Prerequisite_Topics/10.02%3A_Well_Ordering_and_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/08%3A_Some_Prerequisite_Topics/8.02%3A_Well_Ordering_and_Mathematical_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/04%3A_Mathematical_Induction/4.01%3A_The_Principle_of_Mathematical_InductionIn this section, we will learn a new proof technique, called mathematical induction, that is often used to prove statements of the form (∀n∈N)(P(n))
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/07%3A_Sequences_and_Series_Mathematical_Induction_and_the_Binomial_Theorem/7.03%3A_Mathematical_InductionThis section introduces mathematical induction, a proof technique used to establish statements for all natural numbers. It explains the process in two steps: proving a base case and showing that if th...This section introduces mathematical induction, a proof technique used to establish statements for all natural numbers. It explains the process in two steps: proving a base case and showing that if the statement holds for one integer, it also holds for the next. Examples illustrate how to apply induction to verify formulas and inequalities, providing a foundation for proving mathematical statements systematically.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/3%3A_Proof_Techniques/3.6%3A_Mathematical_Induction_-_An_IntroductionConsider P(n):n2+n+11 is prime. In the inductive step, we want to prove that P(k)⇒P(k+1) for ANY k≥1. The following table verifies that it is tru...Consider P(n):n2+n+11 is prime. In the inductive step, we want to prove that P(k)⇒P(k+1) for ANY k≥1. The following table verifies that it is true for 1≤k≤9: n123456789n2+n+111317233141536783101 Nonetheless, when n=10, n2+n+11=121 is composite.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/10%3A_Some_Prerequisite_Topics/10.02%3A_Well_Ordering_and_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_Jeffy_Edits_3.75/09%3A_Sequences_and_the_Binomial_Theorem/9.03%3A_Mathematical_InductionHere we introduce a method of proof, Mathematical Induction, which allows us to prove many of the formulas we have merely motivated previously.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/03%3A_Proof_Techniques/3.05%3A_More_on_Mathematical_InductionBesides identities, we can also use mathematical induction to prove a statement about a positive integer n . Induction can also be used to prove inequalities, which often require more work to finish.
