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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/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/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/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/01%3A_Introduction/1.02%3A_The_Well_Ordering_Principle_and_Mathematical_InductionIn this section, we present three basic tools that will often be used in proving properties of the integers. We start with a very important property of integers called the well ordering principle. We ...In this section, we present three basic tools that will often be used in proving properties of the integers. We start with a very important property of integers called the well ordering principle. We then state what is known as the pigeonhole principle, and then we proceed to present an important method called mathematical induction.
