Categories
Pure Math

Set Theory

What is Set Theory?

Set theory studies sets, which are essentially unordered collections of objects. These objects could be anything from functions to integers to imaginary numbers. Set theory has quite a foundational role in mathematics, as it is helpful background for many different fields in higher mathematics, including group theory, ring theory, and topology. Sets are also very commonly used to signify functions, such as a function f which maps from some input set A to some output set B.

Basic operations and properties of sets

An example of a set is \{1, 2, 4\}, where the numbers 1, 2, and 4 are referred to elements of the set. \{\} denotes the empty set, or the set with no elements, and sets can also be elements of sets. For example, \{\{1, 2\}, \{3, 4\}\} is a set, where its elements are \{1, 2\} and \{3, 4\}.

Set theory introduces the idea of logical operators, such as AND, OR, and NOT. Some helpful concepts to know for set theory are:

  • Union: The union of two sets A and B, written as A \cup B denotes all the elements in A, B, or both. This corresponds to the AND operation.
  • Intersection: A \cap B denotes all the elements in both sets A and B.
  • Complement: A \\ B, or the complement of B in A, denotes all the elements in both sets A which are not in B.

Prerequisites/What I need to know

There are really no prerequisites for set theory, which is what makes it a great place to start in pure math. Some basic background knowledge such as the rational and real numbers as well as of functions is helpful, but not required to learn basic set theory.

Set theory is overall a great foundational subject, but also can be studied as its own subject.

Resources

Skimming these resources should be enough to give a basic background in set theory which can be applied to more advanced endeavours.