Ways to Write Out Sets

4. As we saw on the previous slide, the set rule notation consists of two parts within the curly braces: the part before the $|$ symbol, which is the format of the elements in the set, and the part after the $|$ symbol, which imposes rules on the elements of the set. The $|$ symbol, which is called a 'bar' and is written in $\LaTeX$ as

   $|$

   , is read in English as "such that".
   
   You can include as many rules as you want; also, they can be separated with commas, like in: $\{n\ |\ \text{$n$ is in $\mathbb{P}$, $n > 10$} \}$.
   Here are some examples and their meaning in math and English:
   • $\{n\ |\ \text{$n$ is in $\mathbb{P}$, $n > 10$}\} \;$$= \{11, 12, 13, \dots\}$, "the set of all $n$ such that $n$ is a positive integer, and $n$ is greater than $10$", or "the set of all integers greater than $10$", shortly.
   • $\{n^2\ |\ \text{$n$ is in $\mathbb{N}$}\} \;$$= \{0, 1, 4, 9, 16, \dots\}$, "the set of all numbers $n^2$ such that $n$ is a non-negative integer," or "the set of integer squares", shortly.
   • $\{x^2 + 2x + 1\ |\ $$\text{$x$ is in $\mathbb{R}$}\}$, "the set of all numbers $x^2 + 2x + 1$ such that $x$ is a real number." This is the set of all values that the real quadratic function $x^2 + 2x + 1$ outputs.