Sets

Modern algebra starts from the notion of a set. A set \(S\) is a collection of elements: \[\begin{equation} \require{physics} S = \set{s_1, s_2, \dots, s_n} \thinspace . \end{equation}\] Some examples of sets are the set of natural numbers \(\mathbb{N}\), the set of real numbers \(\mathbb{R}\), the set of complex numbers \(\mathbb{C}\). We can also have smaller sets, for example \[\begin{equation} S = \set{0, 1} \thinspace , \end{equation}\] being the set of the numbers \(0\) and \(1\). Sets don’t necessarily have to contain only numbers. We can, for example, collect all invertible \(n \times n\)-matrices in a set: \[\begin{equation} \text{GL}(n, \mathbb{R}) = \set{A \in \mathbb{R}^{n \times n} ; \text{$A$ is invertible}} \thinspace , \end{equation}\] in which the symbol \(\text{GL}\) has to do with `general linear’, but more on that when we encounter the general linear group.

The previous examples are all concrete (i.e. not abstract) examples of sets. Now say we have a mathematical object, called \(E\) (we haven’t specified anything about it), we can say that \[\begin{equation} G = \set{E} \end{equation}\] is also a set, but in a more abstract sense than the previous examples. We can enlarge this set by adding the elements \(C_2, \sigma_v\) and \(\sigma_v'\), to end up with \[\begin{equation} G = \set{E, C_2, \sigma_v, \sigma_v'} \thinspace , \end{equation}\] in which we still haven’t specified anything about the nature of its elements, but in mathematics that is perfectly fine.