If like me, you’ve spent most of your mathematical high school years introduced to basic sets at the beginning of the year from Grades 8 to 12, then I think you’d agree that sets was one of the quickest and easiest sections we traditionally did. We would quickly recap the same fundamental properties of sets before moving onto more interesting topics, usually algebra. The section would go a little bit like this:

• define the differences between whole and natural numbers, integers, rational numbers and real numbers
• define the differences between unions, intersections and complements, usually through the understanding of Venn-diagrams
• use set builder notation (introducing algebra through this)

If like myself, you truly believed that this was as complicated as sets could ever get, then you, dear reader, like my former-myself, are in for a treat. In university, we build on these basic ideas and have a more in depth understanding about the importance of sets and their greater role in the scheme of mathematics.

My aim is to make these more complex things a bit more accessible and coherent.

As we delve into it, we must first define some essential and familiar terms. $def^n 1$ A set is a collecting of objects called elements.

We denote sets using capital letters and denote their elements with small letters.

To say $\text{set A} = \{x, y, z\}$ contains element x, we write $x\in A$  but $p\notin A$ since p is not an element of set A

These elements are not always abstract values. Elements can be anything such as numbers, points, functions or other sets expressed as a list in brackets {} called braces.

e.g.1. $L = \{\pi, e,\frac{1}{3}, 5, \sqrt{2}\}$ is a set containing 5 real numbers

e.g.2. $\mathbb{Z} = \{...., -3, -2, -1, 0, 1, 2, 3, ....\}$ is the set of all integers (recall: there are infinitely many integers)

e.g.3. $\text{B}= \{(2,3), (4, \pi), (\frac{8}{3}, 0)\}$ is a set containing points on the Cartesian plane (xy-plane)

e.g.4. $\text{F} = \{\text{p, k, l, a, b, u, o}\}$ is a set with 7 letters as elements

e.g.5. $\text{A} = \{\text{thing 1, thing 2, thing 3}\}$ is a set with 3 elements

e.g.6. $\text{N} = \{2, 4, 6, 8, ...\}$ is a set of positive intergers

e.g.7. $\text{J} = \{x \mid x \text{ is positive and even}\}$ is equivalent to set N from e.g.6

e.g.8. $\text{M} = \{\begin{pmatrix} 1 & 9 \\ 8 & 0 \end{pmatrix}, \begin{pmatrix} 7 & 0 \\ 0 & 0 \end{pmatrix}\}$ is a set with two 2×2 matrices

e.g.9. $\text{W} =\{\}$ is the empty set which can also be denoted as $\text{W} = \emptyset$

Looking at the above examples (e.g.6 and e.g.7), we can introduce the concept of  set builder notation.

In general, to write a set X in set builder notation, we write: $\text{X} = \{\text{expression: rule}\}$

Note: The colon is read as “such as” and can also be replaced by a straight vertical line “|”

This means that the expression in the set holds true given it follows certain conditions.

There are multiple ways of writing the same set using set builder notation. For instance, all the sets below are equivalent to $\text{E} = \{..,-4, -2, 0, 2, 4, ....\}$ $\text{E} = \{\text{2n: n is an integer}\}$ which reads “2n exists such that n is an integer” $\text{E} = \{\text{n is an integer : n = 2k, where k is an integer}\}$ which reads “an integer, n, exists given there exists another integer, k, such that n = 2k” $\text{E} = \{\text{n is an integer : n is even}\}$ which reads “n is an integer, given n is even” $def^n 2$: any given set is said to be an infinite set if it has infinitely many elements. If a set is not infinite, then it is finite $L = \{\pi, e,\frac{1}{3}, 5, \sqrt{2}\}$ was an example of a finite set $\text{N} = \{2, 4, 6, 8, ...\}$ was an example of an infinite set $def^n 3$: two sets, A and B, are said to be equal if all the elements in A are also in B

This would mean A is a subset of B, and similarly, B is a subset of A. We will investigate subsets later!

e.g.1. $\{2, 5, 7, 8, 100\}=\{3, 2, 100, 8, 7\}$

e.g.2. $\{.., -1, 0, 1, ...\} = \{0, -1, 1, -2, 2, ...\}$ $def^n 4$: the cardinality of a set refers to the size of the set.

The set must be finite and the number of elements it contains will inform us about its size.

e.g.1. $\text{A} = \{1, 2, 3\}$ has the cardinality of 3 since it has three elements. We denote this as |A|  = 3

e.g.2. $\text{Z} =\{\}$ is the empty set and has the cardinality of 0. We denote this as |Z| = 0. The empty set is the only set that has the cardinality of zero since it does not contain any elements.

e.g.3. $\text{G} = \{1, \{2\},\{3, 4\}\}$ has the cardinality of 3 since it contains three elements, two of which are sets. We denote this as |G| = 3

e.g.4. $\text{W} = \{\{\}, \{\{\}\}, \{\{\{\}\}\}\}$ has the cardinality of 3. We denote this as |W| = 3

e.g.3 and e.g.5 above introduces us to the idea of sets containing other sets.

 How clear is this post?