Set Theory
Set theory gives a precise language for describing collections of objects and reasoning about them.
Imagine describing a large number of objects. It would be tedious to list each object one by one. Instead, we can use set theory to describe the collection of objects in a more concise way.
Suppose we wanted to describe all positive even numbers greater than zero. It would be clumsy to keep referring to the definition. Instead, we could define set $E$ as the set of all positive even numbers greater than zero. Then we can refer to $E$ instead of the definition. This is one of the main motivations for set theory: it allows us to describe collections of objects in a concise way and to reason about them using a precise language.
Set theory is at the heart of all mathematical and scientific research. It is effectively the foundation of all mathematical expressions and scientific reasoning.
Mathematics overcomes language barriers. People that speak different languages from each other can still communicate mathematical ideas quite well. Mathematicians have developed a form of precise symbolic language that allows them to communicate mathematical ideas using symbolic expressions. These symbolic expressions are very concise relative to natural language. The language of mathematics is well defined and unambiguous. The first steps in learning the language of mathematics is learning set theory.
Set theory is the tool used to encapsulate mathematical objects.