What is set theory?
Set theory is the study of discrete structures called sets. What is a set? A set is an unordered collection of distinct items.
For example, the sides on a die can be represented using a set. Or the possible choices — heads and tails — when one flips a coin can be represented using a set. Even the people currently living in the US can be represented using a set and so can the scores in your last three exams.
Speaking of mathematics, all the integers can be represented using a set. Similarly, all real numbers can be represented using a set. You can even have a set of all positive integers and a set of all negative integers. So can you have a set of positive real numbers and negative real numbers.
Significance of set theory
You might be thinking that there's nothing special about set theory; after all, it's just about collecting items in a structure. But that's not true. Set theory isn't only about collecting items together (into a set) but also, and more so, about the ways in which we can process those items.
Set theory is the basis for many other disciplines in mathematics that you shall explore later on, such as functions, relations, sequences, matrices, graph theory, and so on. All these disciplines build on top of the ideas from set theory and, likewise, demand a solid understanding of it.
To surprise you to some extent, efforts have been made to formalize and define entire mathematics on the notion of sets! (Isn't that crazy?) That's the reason it has such foundational significance to mathematics.
Naive set theory
The set theory that is typically studied in a basic course on discrete mathematics is called naive set theory. The reason for using the word "naive" is because it's built around a simple (naive) rule: determine a property and then collect all objects having that property to make a set.
For example, you might suppose the property to be "an integer" — so you collect every single object that is "an integer" to produce a set. This gives you the set of all integers. Or if you suppose the property to be "an athlete", you get the set of all athletes (in the world).
The issue with this supposition is that it can lead to paradoxes, that is, grave inconsistencies. And when a mathematical system falls into the trap of a paradox, it becomes inevitable to redefine it in a way that the paradox is eliminated. Speaking of naive set theory, in order to eliminate such paradoxes, it is redefined in a more formal manner into what's called axiomatic set theory.
Practically, axiomatic set theory is more or less the same as naive set theory, mainly differing on how it's built from the ground up. But it is advanced mathematics and for that reason I won't be covering it in this mini course; instead I'll be covering the standard naive set theory.