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With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. This book begins with a presentation of the rules of logic as used in mathematics where many examples of formal and informal proofs are given. If given numbers \(a\) and \(b\text\) for greatest common divisor.This book gives an introduction to discrete mathematics for beginning undergraduates and starts with a chapter on the rules of mathematical reasoning.
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This helps us understand the structure of the natural numbers and opens the door to many interesting questions and applications.
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In fact, it is a good thing that not every number can be divided by other numbers. This would be going too far, so we will refuse this option. If we wanted to extend our set of numbers so any division would be possible (maybe excluding division by 0) we would need to look at the rational numbers (the set of all numbers which can be written as fractions). Division is the first operation that presents a challenge. If we extend our focus to all integers, then subtraction is also easy (we need the negative numbers so we can subtract any number from any other number, even larger from smaller). It is easy to add and multiply natural numbers.
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Why does 5 and 8 work, 4 and 9 work, but 2 and 4 not work? What is it about these numbers? If I gave you a pair of numbers, could you tell me right away if they would work or not? We will answer these questions, and more, after first investigating some simpler properties of numbers themselves. If we take some number of 2-cent stamps and some number of 4-cent stamps, what can we say about the total? Could it ever be odd? Doesn't look like it. Using this as the inductive case would allow us to prove that any amount of postage greater than 23 cents can be made. In each case we can create one more cent of postage. What if we instead had 4- and 9-cent stamps? Would there be some amount after which all amounts would be possible? Well, again, we could replace two 4-cent stamps with a 9-cent stamp, or three 9-cent stamps with seven 4-cent stamps. You might wonder what would happen if we changed the denomination of the stamps. We were able to prove that any amount greater than 27 cents could be made. Which amounts of postage can be made exactly using just 5-cent and 8-cent stamps? Recall in our study of induction, we asked: What sorts of questions belong to the realm of number theory? Here is a motivating example. It is number theory that makes this possible. Probably the most well known example of this is RSA cryptography, one of the methods used in encrypt data on the internet. This has changed in recent years however, as applications of number theory have been unearthed. Historically, number theory was known as the Queen of Mathematics and was very much a branch of pure mathematics, studied for its own sake instead of as a means to understanding real world applications. This is the main question of number theory: a huge, ancient, complex, and above all, beautiful branch of mathematics. What mathematical discoveries can we make about the natural numbers themselves? Let's take a moment now to inspect that tool. This was the right set of numbers to work with in discrete mathematics because we always dealt with a whole number of things. We have used the natural numbers to solve problems. Section 5.2 Introduction to Number Theory