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Four Fundamental Proof Techniques. Section 1.5 Fri, Sep 9, 2005. Four Proof Techniques. The Principle of Mathematical Induction The Pigeonhole Principle The Diagonalization Principle Proof by Contradiction. The Principle of Mathematical Induction. See Discrete Math.
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Four Fundamental Proof Techniques Section 1.5 Fri, Sep 9, 2005
Four Proof Techniques • The Principle of Mathematical Induction • The Pigeonhole Principle • The Diagonalization Principle • Proof by Contradiction
The Principle of Mathematical Induction • See Discrete Math.
Proof by Contradiction • See Discrete Math.
The Pigeonhole Principle • Let A and B be finite sets with |A| > |B|. Then there does not exist a one-to-one function f : A B. • Thus, if |A| > |B|, then for any function f : A B, it must be the case that f(a1) = f(a2) for some a1, a2 A. • That is, if you have more pigeons than pigeonholes, then at least two pigeons must occupy the same pigeonhole.
Example of the Pigeonhole Principle • Theorem: Let G be a graph with n vertices. If there is a path from vertex A to vertex B (A B), then the shortest path from A to B has length at most n – 1. • (The length of a path is 1 less than the number of vertices along that path.) • Proof: • Let the path be (a0 , a1 , a2, …, am), where a0 = A and am = B, be the shortest path from A to B. • This path has length m.
Example of the Pigeonhole Principle • Suppose that m > n – 1. • Then some vertex ai must be repeated along the path as aj. • That means that the path contains a loop from ai to aj . • This loop may be excised, resulting in a shorter path (a0, …, ai – 1, ai, aj + 1, …, am) from A to B. • That is a contradiction. Therefore, the shortest path must have length no more than n – 1.
The Pigeonhole Principle • The Pigeonhole Principle applies to sets of infinite cardinality as well. • Let A and B be infinite sets with |A| > |B|. Then there does not exist a one-to-one function f : A B. • Similarly, there does not exist an onto function g : B A.
Example of the Diagonalization Principle • Theorem: The set 2N is uncountable. • Proof: • Suppose that 2N is countable. • Then there is a one-to-one function f : 2N N. • That is, the elements of 2N may be listed A0, A1, A2, … • Define the diagonal set D to be D = {i | i Ai}. • Then D Ai for all i. (Why?) • But D is a subset of N, so it must equal some Ai. • That is a contradiction. Therefore, 2N is not countable.
Another Example • Theorem: The set of all functions from N to N is uncountable. • Proof: • Suppose that the set is countable. • Then its elements can be listed f0, f1, f2, … • Define a function g as g(i) = 0 if fi(i) 0, and g(i) = 1 if fi(i) = 0. • Then g fi for all i. • But g is a function from N to N, so it must equal some fi. • That is a contradiction. Therefore, the set is not countable.
Another Example • Theorem: The set of real numbers in the interval [0, 1) is uncountable. • Proof: • Suppose that the set is countable. • Then its elements can be listed x0, x1, x2, … • Write each xiin its decimal expansion: xi= 0.di0di1di2… • Define the i-th digit of a number x to be 0 if dii 0 and 1 if dii = 0. • Then x xi for all i.
Another Example • But x is a real number in [0, 1), so it must equal some xi. • That is a contradiction. Therefore, the set is not countable.
The Diagonalization Principle • What do these three examples have in common?
The Diagonalization Principle • Let R be a binary relation on a set A. • Let D be the diagonal set for R: D = {aA | (a, a) R}. • For each a in A, define a set Ra to be Ra = {b A | (a, b) R}. • Then D is distinct from each Ra.
The Diagonalization Principle • To see that this is true, note that for every a in A, • if aD, then a Ra, and • if a D, then aRa.
The Diagonalization Principle • Adapt this general framework to each of the preceding examples.