Quick Recap of Notation

Here is a list of symbols that we’ve introduced as notation for some of the various ideas that we’ve introduced. Feel free to read it all the way through now, however it’s not meant to be memorized or learned in some deep way. It’s just a collection of symbols, and I want to collect them here so that we can refer to them without having to reintroduce what they mean. If the reader forgets what a symbol means, she or he can come back here. Think of this is a checkpoint, and not as a lesson, because it would be one of the most boring lessons one could possibly write.

In what follows, let $A$ and $B$ be arbitrary sets.

We write $B\subseteq A$ if $B$ is a subset of $A$ . In this case, it is possible for $B$ to equal $A$ . If, however, we want to state that $B$ is a subset of $A$ that is strictly contained in $A$ (i.e., there are some elements in $A$ that are not in $B$ ), then we write $B\subset A$ .

The symbol $\in$ denotes “is in” or “is an element of”. Thus, I can write “ $a\in A$ ” to denote “ $a$ is in A”, or equivalently “a is an element of A”. It’s very important that our mathematics is grammatically correct, and when reading mathematics one should always read it as though it is a normal English sentence.

The following symbols denote important and/or common sets:

$\mathbb{N}=\{1, 2, 3, 4, 5, 6, ...\}$

$\mathbb{Z}=\{...,-3,-2,-1,0,1,2,3,...\}$

$\mathbb{Q}=\{\rm{all\ fractions}\}$

$\mathbb{R}=\{\rm{all\ real\ numbers}\}$

When defining a set, we use “{}” to denote “the set of” and “|” to denote “such that”, as follows. If $A$ is the set of all multiplies of 5 (positive or negative), we could write

$A=\{a\in \mathbb{Z}|a=5\times n \rm{\ for\ some\ } n\in \mathbb{Z}\}$

which we read as

“A is the set of elements “a” in $\mathbb{Z}$ such that a is 5 times n for some n in $\mathbb{Z}$ ”. This is the very definition of being a multiple of 5.

We could then write the set of fractions as

$\mathbb{Q}=\{\frac{a}{b}|a,b\in \mathbb{Z}, b\neq 0\}$

where we take two fractions to be equal if their numerators and denominators are common multiples of each other, in the usual way.

We denote the union of $A$ and $B$ as $A \cup B$ , their intersection as $A\cap B$ , their Cartesian product as $A \times B$ , and their disjoint union as $A \sqcup B$ .
We write $f:A\rightarrow B$ , which is to be read as “f is a function from A to B”.