threenew - Three. The Induction Principle and Properties of...

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Unformatted text preview: Three. The Induction Principle and Properties of Integers 3.1. Three Principles Most (if not all) discrete structures studied in computer science have infinitely many objects, but fortunately, there are usually systematic techniques to investigate the properties about these structures. The single most important such technique is based on an important property of natural numbers known as the induction principle . The set of natural numbers, N = {0, 1, 2, }, consists of all non-negative integers. There are two operations, + (add) and (multiply), defined on N which satisfy the usual algebra laws (closure, commutative, associative, distributive, and identity elements). In addition, and at a more fundamental level, mathematicians have sought after the definition of natural numbers, e.g., what is zero, one, etc. One important outcome of this investigation is the following principle, characterizing the infinite yet simple nature of N : The Induction Principle: Let A N denote a subset that satisfies the following two properties: (1) 0 A ; and (2) if k A , then k + 1 A . Then A = N . Since this is a principle, it is intended to state an obvious fact for which no proof is required. The principle effectively says that starting with the number 0, every natural number can be reached by repeatedly adding 1 for enough times. 3-1, Dr. S. Lang A similar and equivalent principle is the following: The Strong Induction Principle: Let A N denote a subset that satisfies the following two properties: (1) 0 A ; and (2) if 0, 1, , k A , then k + 1 A . Then A = N . Again, this principle should be obviously true. For example, if set A N satisfies the two properties, Property (1) implies 0 A . Applying (2) with k = 0, we conclude 1 A . Then since both 0, 1 A , applying (2) again with k = 1 implies 2 A , etc. Thus, we can conclude A = N . (Of course, this argument is not considered a proof.) There is another (lesser-known) principle of natural numbers, stated as follows: The Well-Ordering Principle: Let B N be a subset of natural numbers, and B . Then B has a smallest element (that is, there is a number s B such that s b for every b B ). The reason that set B in the Principle has a smallest element is that the set of natural numbers N has 0 as its lower bound, preventing its subset B from finding smaller and smaller elements forever. It turns out that any of the preceding principles can be used to prove the others. 3-2, Dr. S. Lang Theorem. The following three principles are equivalent: (a) the Induction Principle; (b) the Strong Induction Principle; and (c) the Well-Ordering Principle. Proof: It suffices to prove (a) (b), (b) (c), and (c) (a). Proof of (a) (b): Let A N and A satisfies the two properties: (1) 0 A ; and (2) if 0, 1, , k A , then k + 1 A . We need to prove A = N . Define C = { m | m N and 0, 1, , m A }....
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threenew - Three. The Induction Principle and Properties of...

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