Ways to Write Out Sets

There are $4$ general ways of writing (expressing) the elements of a set:

1. Listing all the elements (called the roster method,) such as in $\{A, B, C\}$. Good only if the set is finite and short.
2. Using the ellipsis $\dots$ (

   $\dots$

   ) such as in $\{1, 4, 9, 16, \dots\}$ or in $\{1, 2, \dots, 100\}$ for long finite sets or infinite sets.
3. Writing a special list's symbol. Here are some special lists:
   • $\mathbb{N} = \{0, 1, 2, 3, \dots\}$, the set of all non-negative integers. (

     $\mathbb{N}$

     ) ($\mathbb{N}$ is for 'Natural numbers'.)
   • $\mathbb{P} = \{1, 2, 3, \dots\}$, the set of all positive integers. (

     $\mathbb{P}$

     )
   • $\mathbb{Z} = \{\dots, -3, \,$$-2, -1, 0, 1, \,$$2, 3, \dots\}$, the set of all integers. (

     $\mathbb{Z}$

     ) (Zahl means 'whole' in German.)
   • $\mathbb{Q} = \frac{m}{n}$, where $m$ and $n$ are integers: the set of all rational numbers. (

     $\mathbb{Q}$

     ) (From the word 'Quotient'.)
   • $\mathbb{R}$, the set of all real numbers. (

     $\mathbb{R}$

     ), and
   • $\mathbb{C}$, the set of all complex. (

     $\mathbb{C}$

     ).
4. Using a set rule notation (also called the set builder notation) as in $\{n\ |\ \text{$n$ is in}$$\,\text{$\mathbb{P}$ and $n > 10$} \}$, which is read as "all numbers $n$ such that $n$ is in $\mathbb{P}$ and is greater than $10$." More on this on the next slide.