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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/zz%3A_Back_Matter/22%3A_Appendix_B%3A_Mathematical_InductionThere is a variant of one of the bijections we used to prove the Pascal Equation that comes up in counting the subsets of a set. In the next problem, it will help us compute the total number of subsets...There is a variant of one of the bijections we used to prove the Pascal Equation that comes up in counting the subsets of a set. In the next problem, it will help us compute the total number of subsets of a set, regardless of their size. Our main goal in this problem, however, is to introduce some ideas that will lead us to one of the most powerful proof techniques in combinatorics (and many other branches of mathematics), the principle of mathematical induction.
- 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/Monroe_Community_College/MTH_220_Discrete_Math/3%3A_Proof_Techniques/3.6%3A_Mathematical_Induction_-_An_IntroductionConsider \[P(n): \qquad n^2+n+11 \mbox{ is prime}.\] In the inductive step, we want to prove that \[P(k) \Rightarrow P(k+1) \qquad\mbox{ for ANY } k\geq1.\] The following table verifies that it is tru...Consider \[P(n): \qquad n^2+n+11 \mbox{ is prime}.\] In the inductive step, we want to prove that \[P(k) \Rightarrow P(k+1) \qquad\mbox{ for ANY } k\geq1.\] The following table verifies that it is true for \(1\leq k\leq 9\): \[\begin{array}{|*{10}{c|}} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline n^2+n+11 & 13 & 17 & 23 & 31 & 41 & 53 & 67 & 83 & 101 \\ \hline \end{array}\] Nonetheless, when \(n=10\), \(n^2+n+11=121\) is composite.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/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/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/03%3A_Proof_Techniques/3.04%3A_Mathematical_Induction_-_An_IntroductionMathematical induction can be used to prove that an identity is valid for all integers n≥1 .
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/02%3A__Induction_and_Recursion/2.01%3A_Some_Examples_of_Mathematical_IntroductionThe principle of mathematical induction states that in order to prove a statement about an integer n, if we can 1) Prove the statement when n = b, for some fixed integer b, and 2) Show that the tru...The principle of mathematical induction states that in order to prove a statement about an integer n, if we can 1) Prove the statement when n = b, for some fixed integer b, and 2) Show that the truth of the statement for n = k−1 implies the truth of the statement for n = k whenever k > b, then we can conclude the statement is true for all integers n ≥ b.
- https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/01%3A_Tools_for_Analysis/1.03%3A_The_Natural_Numbers_and_Mathematical_InductionWe will assume familiarity with the set N of natural numbers, with the usual arithmetic operations of addition and multiplication on n , and with the notion of what it means for one natural number ...We will assume familiarity with the set N of natural numbers, with the usual arithmetic operations of addition and multiplication on n , and with the notion of what it means for one natural number to be less than another.
