In this article, I will introduce you to sets and its notations. There are many definitions for sets. But no one single definition is acceptable. So, I will start with a simple informal definition of a set.
“A set is group of objects with some properties and sometimes the object from same group does not share same properties”.
![Figure 1 - Set A is a set of students from a school.](https://notesformsc.org/wp-content/uploads/2022/12/Figure1-SetA-set-of-students.png)
Suppose is a set of students from a school. Students in school are a in a set, but all students do not study same course. Some study math and other study programming and so on. In other words, they have different properties.
![Figure 2 - Set of fruits with different tastes and colors](https://notesformsc.org/wp-content/uploads/2022/12/Figure-2-Set-of-Fruits.png)
Let be a set of fruits: it has
and all fruits have different taste and color.
What is the definition of set?
The formal definition of a set is:
A set is an unordered collection of objects called its members or elements. A set contains elements or members.
Here unordered is important and we will come back to it later. If an element belongs to a set, then we write which means element ‘a’ is in set A.
![Figure 3 - Element a is in set A is shown with the "in" operator.](https://notesformsc.org/wp-content/uploads/2022/12/Figure-3-Element-in-set-A.png)
If an element does not belong to a set, we write which means that element ‘a’ is not in set A.
![Figure 4 - Symbol to indicate that element a is not in set A.](https://notesformsc.org/wp-content/uploads/2022/12/Figure4-element-not-in-set-A.png)
Set Notations
There are two ways to describe a set.
- Roster notation
- Set builder notation.
Roster notation
In roster method, you simply list all the elements of a set within braces. If is a set of soft drinks, then list all the elements inside curly braces, separated with a comma. You have already seen an example above.
![Figure 5- Roster notation for sets](https://notesformsc.org/wp-content/uploads/2022/12/Figure5-Roster-notation.png)
The roster method works when you have limited elements in a set. In other words you can count the number of elements in a set.
Set Builder Notation
When the size of set is too large, and you are not able to list the element, use the set builder notation. You are simple creating a rule for the elements of a set.
For example:
Suppose is set of all numbers less than
, then you write this in set builder notation within curly braces like this.
A = {x \in N \mid x < 100}
![Figure 6 - Set builder notation](https://notesformsc.org/wp-content/uploads/2022/12/Figure-6-Set-builder.png)
Note that we wrote as a set, not
because
is set of natural numbers from which we got elements of
. Sometimes elements of a set come from a bigger group called its Universe.