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Notating Sets: How we write domains, codomains, and ranges precisely.

An important piece of mathematical language is properly writing sets, and there are a number of ways to do this. One such way, which we will outline below, is the so-called ‘set-builder notation’. This is essentially building a set by listing the properties that each member (or “element”) of the set has. Consider the following statement:

$$\{x : x \text { is a drink in the vending machine} \}$$ The above statement has a number of very important symbols. ^[1] ( I will describe them more formally, but then revise what the above set says in English to try and clarify. You may need/want to read this part a few times. If you don’t understand set notation you have a near-zero chance of passing this - or any future mathematics - course, so take the time now as it will be worth it in the long run. ) The braces on either end signal the beginning and end for the set. Essentially the braces are saying ‘this is a set/collection of things’. The variable $x$ in the first part (before the colon) is a generic placeholder for something in the set - often called a “dummy variable”. The colon is a delimiter that transitions from the notation for the generic representative to the section outlining properties of elements of the set (the colon is usually translated as “such that”). Finally, the content after the colon is a (comma-separated) list of traits that something must have in order to be in the set.

That may seem pretty dense and hard to understand, but it helps to translate the set that we wrote into English. That set, if you translate it literally, says: ‘This is the set of things (which we will call $x$ for a second), such that each of those $x$ is a drink in the vending machine’. In a more human translation you would say that ‘this is the set of drinks in the vending machine’.

There are a number of commonly used symbols and notations that we list next with description.

$\mathbb {N}$ :

This symbol is the set of all natural numbers. Specifically it is the numbers $\{1, 2, 3, \dots \}$, ie all strictly positive integers. Note that, in some courses, the natural numbers may include zero, but not in this course (this is, weirdly, a rather hotly debated topic among some mathematicians).

$\mathbb {Z}$ :

This symbol is the set of all integers. Specifically it is the numbers $\{\dots , -3, -2 ,-1, 0, 1, 2, 3, \dots \}$. These are all the positive and negative whole numbers, along with zero.

$\mathbb {Q}$ :

This symbol is the set of all rational numbers. Specifically, all fractions that have integers for their numerator and denominator.

$\mathbb {R}$ :

This symbol is the set of all real numbers. Specifically, it includes all numbers that do not include the imaginary unit $i$.

$\mathbb {C}$ :

This symbol is the set of all complex-valued numbers. Specifically it includes all real numbers and all products of any real number with the imaginary unit $i$.

$\in$ :

This symbol is translated as “in” or “is an element/member of”. For example, you could see $7 \in \mathbb {N}$, which should be read as “$7$ is an element of the natural numbers.

$\emptyset$ :

This symbol is the “empty set”. Specifically it is the set with nothing in it; which is different from “$0$”.