1.1: The Set of Natural Numbers
The first mathematics we learned as children was how to count: “One, two, three. . .” (or “Un, deux, trois. . .” or what have you). Only later did we learn about other kinds of numbers: fractions and negative numbers, for example. The counting numbers are thus the most “natural”, and the set is sometimes called the set of natural numbers. However, for certain technical reasons it’s convenient to include zero and so we define the set of natural numbers as .
The set brackets “” and “” tell us to consider the collection of things between them as one object—namely a set. The symbol “” is just an abbreviation to allow us to refer to this object concisely. The elements of a set are the “things” in the collection. The symbol “” is pronounced “is an element of”, so for example is an abbreviation for the sentence “Thirty-seven is an element of the set of natural numbers.”.
The ellipses ( … ) in the definition of tell us to continue the pattern displayed. The result is an infinite set—one with no last element. Ellipses are also sometimes used in defining finite sets, for example the familiar
The usual convention is to use capital letters to refer to sets. To read the definition out loud, you could say, “ is the set whose elements are zero, one, two, and so on forever.” The formal way to pronounce the symbol is “the set of natural numbers”. Usually, unless there is some need to be precise, one simply says “en”.
This can be trickier than it might appear because it’s common to use “n” as a variable. The convention here is that lower-case letters refer to numbers. So one can say, for example, “If , then must be either even or odd”. This should be pronounced “If en is a natural number, then …”.
The slash “/” is used to denote negation, as in . This idea is common even outside of mathematics, where we find “” (meaning “no”) on street signs, for example. So it shouldn’t come as any surprise that stands for the (true) sentence “One-half is not an element of the set of natural numbers.” (or just “one-half isn’t in ” or some such variation).
There are several other common ways to say “”: for example, “Two doesn’t go into one” or “One is not a multiple of two”. Here is another way: This is pronounced “Two does not divide one”. In general, whenever we have natural numbers n and m we can say that n divides m, and write , whenever . We can also say in this case that n is a factor of m, or that n is a divisor of m.
Thus, for example, , because and . This can be rephrased slightly as “because and 19 is a natural number”. But because there is no natural number with . In other words, because .
Two special cases involve zero. Suppose we have and . Then and To confirm these facts, look at the definition of “”: since and , we get as advertised. But . In fact isn’t a number at all! Be careful when working with 0 to avoid thinking of the wrong special case.
In the original version [~1995] of this piece, several exercises follow (of course!). As of this moment, they’re too much trouble to reproduce here. Exercise: find an expert user of the various notations and discuss.