In Mathematics, a set is an unordered collection of objects, known as elements or members of the set. It is widely used in many branches of Mathematics such as Number Theory. The standard notation for a set is the brace notation \(\{ \cdots \}\), with all elements of the set listed inside the pair of braces, for example, \(A=\{2, 4, 6, 8\}\). The second method to build a set is to use the brace notation as \(\{ \cdots \mid \cdots \}\) where the vertical bar “\(\mid\)” within the brace notation means “such that”, the dots to the left of the bar represent a generic form of the element, while the dots to the right of the bar provide any constraints on the element, for example,

\[R=\{ x \mid 0 \le x \le 1 \}\]

denotes all the numbers between \(0\) and \(1\) including \(0\) and \(1\). In some textbooks, the vertical bar is replaced by colon (\(\colon\)) such as
A third method is to use semicolon (;) for colon. This is not recommended.

\[R=\{ x : 0 \le x \le 1\}\]

Both notations are well accepted. In this course, we will be using the first notation with the vertical bar.

Special Sets

In Number Theory, there are many special sets of numbers defined by using the blackboard bold typeface. Special sets of numbers include

\(\mathbb{R}\): the set of all real numbers.

\(\mathbb{N}\): the set of all natural numbers.

\(\mathbb{Z}\): the set of all integers including zero, positive, and negative numbers (from the German word Zahl).

\(\varnothing\): an empty set (a set with no elements).

With that, we can now define the set \(R\) more precisely as

\[R=\{ x \in \mathbb{R} \mid 0 \le x \le 1 \}\]

which now represents all the real numbers between \(0\) and \(1\) including \(0\) and \(1\).