Introduction to Sets

The idea of a set is to represent, refer to, and use a collection of objects by treating the set as a single entity, which means a single mathematical object. The objects in such a collection are said to be its elements or members. If a collection of distinct elements is well-defined“Well defined” means that the criteria to determine whether a particular object is in the collection is crisp and unambiguous. and there’s no order among its members, we call it a set.

A set is represented using a capital letter, whereas a small letter is used to represent the members of the set. For example, let’s consider the following set:

The set $S$ is defined to contain five members shown explicitly enclosed in curly brackets and separated by commas. Sometimes, the members of the set are explicitly listed, as is the case in the following example:

Another essential characteristic of a set is that it must contain no repetition or order of its elements. Therefore, we should feel comfortable with the following:

To represent that an element $e$ is in the set $S$ or is an element of $S$, we write $e\in S$. Sometimes, we also read it as $e$ is a member of the set $S$. From the examples given above, we can write that $o\in S$ and $\text{guava}\in F$. If we want to say that $g$ is not an element of the set $S$, we write it as $g\not\in S$.

Let’s see a few of the most used methods to describe a set.

Tabular notation

Set members are separated by commas in tabular notation and listed within a pair of curly brackets. In the example above, the set $S$ is defined using tabular notation. Here are a few examples of set definitions in tabular notation:

Set-builder notation

As we have seen in the examples above, each example set has a fixed number of elements. Such sets are called finite sets. On the other hand, a set might contain infinitely many elements. A set that isn’t finite is said to be an infinite set. For instance, $\{1, 2, 3, 4, 5, \ldots\}$ ...