Re-introduction to Mathematics: Sets and Notation (Pt. 0)

Mathematics is the foundation of the world around us, it shapes the way we reason, prove, and explore new ideas.

This article is the first in a series where we’ll break down essential topics that underpin higher-level mathematics. Our journey begins with sets, exploring how they are defined, the notation used to describe them, and the basic operations we can perform with them.

Whether you’re brushing up on mathematical fundamentals or encountering these ideas for the first time, this series aims to provide clarity, motivation and intuition, to hopefully make some abstract concepts more approachable.

There is very little prerequisite knowledge required. If you can count add and multiply, you have all the skills you’ll need.

If you’re joining from a background in high school level math, I hope this is a better experience than you are perhaps used to. There will be no formulae to memorise, no exercises to complete, and it should feel very different to the style of math you may be used to.

Let’s begin by understanding what a set is and why it’s such a powerful tool in mathematics.

Introducing the Set

In math, we often talk about collections of things. A set is just that: a collection of things.

The elements of a set could be anything: numbers, equations, lines, shapes, fruit or any other thing you can dream up. Let’s be a little more specific and give a more precise definition of a set.

A set is a well-defined, unordered, distinct collection of objects.

• well-defined: it should be absolutely clear what elements do and don’t belong in a set, it cannot be ambiguous,
• unordered: the elements are not ordered based on their value, and they are not assigned a position within the set,
• distinct: there can be no duplicate elements within a set, all elements must be unique.

Sets are a fundamental concept and understanding their notation is crucial to navigating the world of mathematics. Let’s see some examples:

We can construct a set containing the numbers 1 to 4:

\[\{ 1, 2, 3, 4\} \]

Since elements are unordered, we could also write \(\{4, 3, 2, 1\}\) and \(\{3, 1, 4, 2\}\). In fact, all three of these sets are identical.

It also doesn’t make sense to refer to the ‘first’ element of a set. They don’t have an order so we can’t say that one comes before or after another.

You’ll also notice that there are also no duplicate elements in the set above. As a counterexample, \( \{1, 1, 2, 3, 4\}\) is not a valid set. Instead, this would be referred to as a multiset or a bag, but we don’t often use these types of objects. It turns out uniqueness is a very useful property of a set that comes up a lot in mathematics.

We can also create a set with no elements. This is known as the empty set and has its own symbol, \(\emptyset\).

As we mentioned before, sets aren’t limited to containing just numbers, even if that is how they are most commonly used. We could, if we wanted, even have a set whose elements are themselves sets – a set of sets!

\[\{\{ 1, 2, 3\} , \{ 4, 5, 6\} , \{ 7, 8, 9\}\}\]

An element of…

To indicate that an element belongs in a set, we have the following notation

\[ 2\in\{1, 2, 3\}\]

This should be read as:

The element, 2, is an element of the set, \(\{1, 2, 3\}\)

with the symbol \(\in\) translating to ‘is an element of’ or simply, ‘in’.

Sometimes we use this same notation to indicate that we are choosing an element from a set. Writing

\[x\in\{1, 2, 3, 4\}\]

would mean that we are choosing an element at random from the given set. This is useful when writing a mathematical proof as it allows us to prove something for all elements of a set in one go.

More Complex (and more useful) Set Definitions

Sometimes it’s not practical to list all of the elements that belong in a set.

If the range of numbers in a set is too large to reasonably write down, we can instead write it like this:

\[\{ 1, 2, \dots, 10\}\]

This doesn’t just have to be used for consecutive numbers, we could write something like:

\[\{2, 4, \dots , 10\}\]

As long as it’s clear what the pattern is and there is no ambiguity about what the set contains.

We can also use simple formulas to describe the elements of a set:

\[\{x \ |\ x+2 \ge 5 \} = \{ 3, 4, 5, \dots\}\]

denoting the set of whole numbers greater than 3. This should be read as:

The set containing elements, x , such that \(x+2 \ge 5\)

In reality, we’d probably write this in a simpler form: \(\{ x | x \ge 3\}\). There’s no need to overcomplicate things!

The symbol, ‘ | ‘, roughly translates to ‘such that’. You may also see it written using a \(:\), e.g. \(\{ x : x \ge 3\}\)

Using our notation from above, we can formulate a more complicated set like this:

\[ S = \{ x^2 | x\in \{1, 2, 3\}\]

…the set of elements of the set \( \{1, 2, 3\} \) each squared, producing the resulting set, \( \{1, 4, 9\} \)

Common Sets

There are some sets of numbers that are very commonly used, so we have special notation to refer to them. We have:

• \(\mathbb{Z}= \{\dots, -3,-2, -1, 0, 1, 2, 3, \dots \}\), the set of all integers
• \(\mathbb{N}= \{1, 2, 3, \dots \}\), the set of all positive integers, known as the natural numbers
• \(\mathbb{Q}= \{\frac{a}{b} | a, b \in \mathbb{Z} \}\), the set of all fractions, known as the rational numbers
• \(\mathbb{R}\), the set of all real numbers. This includes all positive and negative numbers, all numbers that can be written as fractions, and all irrational numbers (\(\pi ,\sqrt{2} \))

You’ll see these sets being referenced frequently in papers, textbooks and online proofs.

Set Operations

We can perform operations with sets. Let’s say we have 2 sets, A, and B.

\[A = \{1, 2, 4, 5, 7\}, B = \{3, 4, 5, 6, 8\}\]

We can take the intersection of these 2 sets, \(A \cap B\), which produces a set containing the elements common to both A, and B.

\[A \cap B = \{ x | x\in A \ and\ x \in B\} = \{4, 5\} \]

We can also find the union of A and B. This produces a set containing elements that are in at least one of A or B (or both). Essentially, this operation combines all elements of both sets together while removing any duplicates.

\[A \cup B = \{ x | x\in A \ or\ x \in B\} = \{1, 2, 3, 4, 5, 6, 7, 8\} \]

We can also find the difference of two sets (sometimes called the set-theoretic difference). This involves removing all elements of one set from another. For example, \(A/B\), would result in the set A with any elements that appear in set B removed from it.

Let \(A=\{1, 2, 3, 4, 5, 6\}\) and \(B=\{1, 2\}\). Then \(A/B = \{3, 4, 5, 6\}\) and \(B/A = \emptyset\).

The last, however less common, set operation you may encounter is the Cartesian product, \(\times\). To understand the cartesian product, we first need one more definition. A tuple is similar to a set, except without the constraint that elements are unique, and we now assign an order to the elements. The classic example of this is a coordinate point e.g. \((3, 4)\).

If we relax the condition of a set having no duplicates and assign an order to its elements, we get an object known as a tuple. An example of this is a set of coordinates. The coordinate, \((3, 4)\) is a tuple of length 2. The order of the elements does matter, as \((3, 4)\) is not the same as \((4, 3)\).

Performing a Cartesian product operation produces a set of tuples of length 2. In fact it produces a set containing all possible combinations of length 2 tuples where the first element is from one set, and the second element is from the other. These 2-tuples are sometimes referred to as ordered pairs, and may be written using \(<, >\) instead of regular brackets.

Let A and B be sets, The Cartesian product of A and B, \(A\times B\), produces the set \(\{<a, b> | a\in A, b \in B\}\).

This operation is a little more confusing, and is easier to understand with an example:

Let \(A=\{1, 2, 3\}\) and \(B=\{4, 5\}\). Then \(A\times /B = \{<1, 4>, <1, 5>, <2, 4>, <2, 5>, <3, 4>, <3, 5>\}\)

Each element of the resulting set has it’s first element belonging to A and its second element belonging to B.

Subsets and Supersets

The final definitions we will explore is the notion of a subset. We say that A is a subset of B, \(A\subset B\) if every element of A is also an element of B.

For example, let \(A=\{1, 2, 3\}\) and \(B=\{1, 2, 3, 4, 5 \}\). Then A is a subset of B, \( A\subset B\).

We also have a name to refer to this relationship from the opposite perspective. Instead of saying A is a subset of B, we could also say that B is a superset of A, meaning that B contains all the elements of A within it.

It’s actually true that every set is a subset of itself!

\[ A\subset A \]

To convince yourself, revisit the definition of a subset: all the elements of A are also elements of A.