Sets

We use {} to denote a set and the set {} that has no elements is commonly referred to as the empty set or the null set and may also be written $\emptyset$. The set {a, b} contains two unique elements a and b. The set {a, a, a, a, b} is identical to {a, b} or {b, a}. Sets do not care about the order in which they are listed or repeated entries (hence the emphasis on unique elements).

Each member of a set is generally called an element. There are some common sets that will recur throughout all of your mathematical education and the have shorthand double-struck notation- sometimes called blackboard font. We use … (an ellipsis) to indicate that a set goes on forever in the obvious pattern. For example, {1, 2, 3, …} is the set of counting numbers or natural numbers and has the double struck abbreviation $\mathbb{N}$.

In some sense, the primary property of a set is the number of elements in the set. The secondary property is the type of element. We denote the size of a set with the term cardinality and use absolute value bars.

$$|\{a , b, c\}| = 3$$

$$\text{The set \{a, b, c\} has cardinality 3}.$$

If a set has an infinite number of members, we say the set has infinite cardinality (we will revisit this later in more depth).

We say two sets are equivalent if they have the same number of elements. We use the symbol $\cong$ to denote this. Two sets are equal if they contain exactly the same elements (though not necessarily in the same order).

Set theoretic notation is convenient and universal among mathematicians. There are some subtle variations and choices to be made, but with a basic understanding of set notation, you will easily understand any nuances in context.

This section will focus on two key aspects of mathematics: modern mathematical notation and the construction of sets.

There are a variety of symbols in common usage in modern mathematics. We will review most of the common ones and I will give you both the original definition of the symbol and the modern English interpretation. We’ll go over these in some detail and you’ll see them often repeated in this and all future courses. You don’t need to spend a lot of time trying to memorize them now, it will come naturally through exposure. Additionally, English has largely become the lingua franca for math and science. As a result, we tend to only use the right facing symbols for inclusion (like element of or subset of). The other symbols have taken on a new meaning. A great example of this is that it is rare to see element of a set written left face as $\ni$, but it has become common to combine it with the : in the following way $\ni :$ to read “such that “.

Common Symbols

$\exists$	There exists	There is a
$\forall$	For all	For every
$!$	unique	only one
$\in$	is an element of	is in
$:$ or $\ni :$	such that	such that
$\{ \text{stuff in here} \}$	the set of (stuff)	the set of (stuff)
$\subset$	subset of	is part of
Strike through means not, for instance $\notin$	is not an element of	not in
$a_1, a_2, a_3$	read “a 1, a 2, a 3”	indicates that generic elements that are different
$\infty$	infinity	infinity
$x \rightarrow y$	x implies y	if x is true then y is true
$x \longleftrightarrow y$	x implies y and y implies x	if x is true then y is true and if y is true then x is true

Common Sets

$\mathbb{N}$	the set of Natural Numbers	{1, 2, 3, …}
$\mathbb{Z}$	the set of Integers	{…, -2, -1, 0, 1, 2, …}
$\mathbb{Q}$	the set of Rational Numbers	$\{\frac{p}{q} \ni : p, q \in \mathbb{Z}, q\neq 0\}$
$\mathbb{R}$	the set of Real Numbers	we will clearly define this later!
$\mathbb{C}$	the set of Complex Numbers	$\{a+bi \ni : a, b \in \mathbb{R}\}$

Using our knowledge from the last section, we may write some useful and interesting examples:

• $a \in \{a, b\} \text{which is read: a is an element of the set \{a, b\}}$
• ${a, b} \subset \{a, b, c\} \text{ which is read: the set \{a, b\} is a subset of the set \{a, b, c\}}$
• $\{1, 2, 3, … \} = \mathbb{N} \text{ which is read: the set 1, 2, 3, and so forth is identical to the set of natural numbers}$

Let’s try to go the other way now.

We also tend to use symbols like union $\cup$ and intersection $\cap$ to capture sets more fully. One of the common ways in which we do this is something known as interval notation. [0, 1] is an interval that includes 0 and 1 and all of the real numbers in between the two. This is typically called a close interval. (0, 1) is the same interval except for the endpoints 0 and 1. This is typically called an open interval. We also have the possibility of half open intervals like (0, 1]. This is a common way to notate subsets of the real numbers for a variety of different methods.

Some sets are harder to describe in English than in math. For instance: all the rational numbers strictly between 0 and 1. We can notate this like $\mathbb{Q}\cap(0,1)$.

For instance, we may want to talk about the domain of a function such as $f(x) = \frac{1}{x}$. We typically imagine the domain is the largest of set of real numbers that allow for computation of sensible values of f(x). Here, we see that we cannot divide by 0 and so we consider the domain to be $(-\infty, 0)\cup(0, \infty)$.

We use slightly different notation to denote set containment. Let’s call our set $A = \{-3, -2, -1, 0, 1, 2, 3, 4\}$. We say $4 \in A$ or 4 is an element of the set A, but we could say something that is slightly different. We could say $\{4\} \subset A$ which is the set containing 4 is a subset of A. There is some disagreement with the use of the symbols $\subset$ and $\subseteq$. Often the second symbol is used when it is possible the subset is the whole set, but it has become common to use these symbols interchangeably.

There is a valid reason for this confusion as well; In general, when I specify a set it can be difficult to know precisely what is included or even the cardinality of the set. For instance, $|\mathbb{Z}| = \infty$ and $|\mathbb{Q}| = \infty$, but $\mathbb{Z} \subset \mathbb{Q}$ and, in fact, $\mathbb{Z}$ is a proper subset! A proper subset is a subset that contains fewer elements than are in the larger set. Note that for infinite sets, cardinality is insufficient to determine whether a subset is proper or not. For finite sets, it is enough.