We use special capital letters to make this notation easy to recognize: \(\A\text{,}\)\(\N\text{,}\)\(\PP\text{,}\)\(\W\text{,}\) and \(\Z\text{.}\) These letters are written in a font called blackboard bold. Some sets in the following definition are already familiar. We will see why the others are important later.
We have already seen the integers and natural numbers in Part I and now have a handy way of talking about all of these as the set of integers \(\Z\) and the set of natural numbers \(\N\text{.}\) The set \(W\) contains all elements of \(\N\) and also the number \(0\text{.}\)
We will use the sets \(\Z_n\) and \(\Z_n^\otimes\) throughout this course. The set \(\Z_n\) contains exactly the numbers that can occur as remainders when dividing by \(n\text{,}\) that is
We use the set of characters \(\A\) to encode characters as numbers in Section 8.1 and to encode texts as numbers in Section 12.3. We will also use it when encrypting messages in Section 8.3, Section 8.4, and Section 16.3. To distinguish characters from variables we write characters in a typewriter font.
In Figure 5.4.2 we show sets from Definition 5.4.1 on the number line. For the sets \(\Z_n\) and \(\Z_n^\otimes\) we chose the special case \(n=7\text{.}\) In Example 5.4.3 we consider the sets \(\Z_n\) and \(\Z_n^\otimes\text{,}\) where \(n = 6\text{.}\)
It is useful to check what a definition means in the borderline cases. The sets \(\Z_n\) and \(\Z_n^\otimes\) are defined for all natural numbers \(n\text{.}\) We check the definition for the two lowest allowed values of \(n\text{,}\) namely \(n=1\) and \(n=2\text{.}\)
The video in Figure 5.4.5 reviews the definitions of special sets. It also shows how to write statements from Chapter 1 in a shorter way by using set notation.
We can use set notation and the special sets defined above to give shorter formulations of statements from Section 1.3. Essentially we are replacing “let \(a\) be an integer” by “let \(a\in\Z\)”.
