《Discrete Mathematic with Applications》读书笔记四

Chapter 4 SEQUENCES AND MATHEMATICAL INDUCTION

One of the most important task of mathematics is to discover and characterize regular patterns, such as those associated with processes that are repeated.

The main mathematical structure used to study repeated processes is the sequence, with mathematical induction.

4.1 Sequences

Summation Notation

When the upper limit of a summation is variable, an ellipsis is used to write the summation in expanded form.

Product Notation

Factorial Notation

Properties of Summations and Products

Proof is discussed in Section 8.4

Change of Variable

Sequences In Computer Programming

One-dimensional arrays to presents a finite sequences.

Application: Algorithm to Convert from Base 10 to Base 2 Using Repeated Division by 2

4.2 Mathematical Induction I

Mathematical Induction is used to check conjectures about the outcomes of processes that occur repeatedly and according to definte patterns.

The validity of proof by mathematical induction is generally taken as an axiom.

Below is a formal version of the proof about coins developed.

This derivation of the fomula is attractive and is quite convincing. However, it is not as logically airtight as the proof by mathematical induction. To go from one step to another i the calculations above, the argument is made that each term among those indicated by the elipsis(...)has such-and-such and appearance and when there are canceled such-and-such occurs. But it is impossible actually to see each such term and each such calculation, and so the accuracy of these claims cannot be fully checked. With mathematical induction it is possible to focus exactly on what happens in the middle of the ellipsis and verify without doubt that the calculations are correct.

4.3 Mathematical Induction II

We will use the mathematical induction to prove additional kinds of statements such as divisibility properties of  the integers and inequalities.

Proving a Divisibility Property.

Proving an Inequality

for the second line of the proof. There is a error for the original inequality. The correct one shoule be : 2n+1 < 2^n .

The last example demonstrates that how to use mathematical induction to show that the terms of a sequence satisfy a certain explicit formula.

4.4 Strong Mathematical Induction and the Well-Ordering Principle

The strong mathematical Induction is a technique for establishing the truth of a sequence of statements about integers.

All statement can be proved with ordinary mathematical induction can be proved with strong mathemtaical induction.  And it is the same for the strong mathemtaical induction.

The strong mathematical induction can be written without a basis step if the inductive step is changed to V k >= a, if p(k) is true then p(k+1) is true.

Applying Strong Mathematical Induction

Binary Representation of Integers

If smaller integers than n have unique representations as sums of powers of 2, then the unique representation for n as a sum of powers of 2 can be found by taking the representation for n/2 (or for (n-1)/2 if n is odd) and multiplying by it by 2

The Well-Ordering Principle for the Integers

The ordinary and the strong mathematical induction, and the well-ordering principle are equivalent.

4.5 Application:Correctnes of Algorithms

Assertion:

Predicate scribing the initial state is called the pre-condition for the algorithm and the predicate describing the final state is caled the post-condition for the algorithm.

Loop Invariants

The two other example for the correctness of division algorithm and euclidean algorithm ref to the textbook.