Set Theory

Sets are one of the most fundamental building blocks in mathematics. Set Theory is the study of sets of objects, and the relationships between them. For example, $$ \{1,2,3\} $$ is the set of the numbers $1$, $2$, and $3$. The curly braces $\{\}$ are used to denote a set, and the elements of the set are listed inside, separated by commas. And while we did put numbers in this set, we could’ve put anything we wanted, like symbols: $$ \{\lambda,\star,\square,\Sigma\} $$

or, the set of all of Pluto’s moons: $$ \{\text{Charon},\text{Nix},\text{Hydra},\text{Kerberos},\text{Styx}\} $$

The Naturals and The Integers

While we could sit around and make any set we wish, there are a handful of sets that are fundamental, already familiar, and extremely useful.

The natural numbers are the set of all positive whole numbers–the “counting numbers” that we use to count objects: $0,1,2,3,\ldots$ and so on.

The integers are the set of all whole numbers, both positive and negative, and including zero: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ and so on.

These sets are so fundamental that they have their own special symbols: $$ \begin{aligned} \mathbb N&=\{0,1,2,3,\ldots\} \\ \mathbb Z&=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\} \end{aligned} $$

A Note About Notation

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There is no consensus on whether or not $0$ is included in the natural numbers–it’s like whether you think Pluto is a planet or not. I think those in the camp of planethood would say that $0$ is a natural number, and so I include it in $\mathbb N$. Sometimes this will be made explicit by writing $\mathbb N_0$. If $0$ is not to be included then you may see it written as $\mathbb N_1,\mathbb N^+$, or $\mathbb N^*$. However, I will strictly be using $\mathbb N$ generically, and you can just assume that it includes $0$ (I’m a planethood for Pluto kinda guy).

Important Concepts and Notation

Set theory is a load-bearing foundation for all of mathematics, so it would be good to touch on some stuff before going on to the main event. We’ll need these tools later.

The simplest is the idea of an element being in a set. We write $x\in S$ to mean: “$x$ is an element of the set $S$”, or more simply, “$x$ is in the set $S$”. This is referred to as set membership. Often times, we want to talk about the size of a set–how many elements are in the set–which is called the size/order/cardinality of the set. We denote this as $|S|$.

As pure mathematicians go, we must prove things about the systems we study. In this vein we often like to make provable claims about all elements of a set. This is really just saying, “for all $x$ in the set $S$, some property $P$ holds”. To express these ideas, we use $\forall x$ to mean “for all $x$”, and $P(x)$ to mean “the property $P$ holds for $x$”. So, putting it all together, we can write: $$ \forall x\in S, P(x) $$

Additionally, if $P(x)$ does not hold for $x$, we can write $\neg P(x)$, which is read as “not $P(x)$”.

The final two symbols we’ll need are $\exists$ and $\mid$. The symbol $\exists$ means “there exists” – so $\exists x\in S$ means “there exists an element $x$ in the set $S$”. This symbol is then often combined with $\mid$ in the following manner:

$$ \exists x\in S\mid P(x), $$ which is to be read as “there exists an element $x$ in $S$ such that the property $P$ holds for $x$”.

Alright, with all the basic set notation out of the way, we can move up the abstraction ladder and get to the Algebra part of Abstract Algebra. Here’s a quick summary of the notation we just covered: