We won’t make this section of the text too long \u2014 all we really want to do here is to take a short memory-jogging excursion through little bits and pieces you should remember about sets and numbers. The material in this chapter will not be (directly) examined.<\/p>\n
Before we do anything else, it is very important that we agree on the definitions and names of some important collections of numbers.<\/p>\n
Natural numbers \u2014 These are the \u201cwhole numbers\u201d 1,2,3,\u2026 that we learn first at about the same time as we learn the alphabet<\/a>. We will denote this collection of numbers by the symbol \u201c \u201d. The symbol is written in a type of bold-face font that we call \u201cblack-board bold\u201d (and is definitely not the same symbol as ). You should become used to writing a few letters in this way since it is typically used to denote collections of important numbers. Unfortunately there is often some confusion as to whether or not zero should be included . In this text the natural numbers does not include zero.<\/p>\n Notice that the set of natural numbers is closed under addition<\/a> and multiplication<\/a>. This means that if you take any two natural numbers and add them you get another natural number. Similarly if you take any two natural numbers and multiply them you get another natural number. However the set is not closed under subtraction or division; we need negative numbers and fractions to make collections of numbers closed under subtraction and division.<\/p>\n Two important subsets of natural numbers are:<\/p>\n All of you will have done some basic bits of set-theory in school. Sets, intersection, unions, Venn diagrams etc etc. Set theory now appears so thoroughly throughout mathematics that it is difficult to imagine how Mathematics could have existed without it. It is really quite surprising that set theory is a much newer part of mathematics than calculus. Mathematically rigorous set theory was really only developed in the 19th Century \u2014 primarily by Georg Cantor . Mathematicians were using sets before then (of course), however they were doing so without defining things too rigorously and formally.<\/p>\n","protected":false},"excerpt":{"rendered":"\n
Sets<\/h3>\n