Definition

What is a set? One may think of a set as a collection of objects. These objects can be anything: books, numbers, words. Typically, the objects have some kind of relationship with each other. For example:

1. The set of all books written by Kurt Vonnegut
2. The set of all positive numbers on the real line
3. The set of all words on this page

These are only a few examples but one can start to see our motivation for defining a set. The objects that make up sets are elements. We say an element a belongs to a set S, denoted as:

We can also say an element a does not belong to S, denoted:

Let’s say A₁, …, A_n are objects. Then the set of these objects is {A₁, …, A_n}. We use curly braces { } to define sets.

Sets in statistics and probability

Sets are also an important element in probability and statistics. The mathematical framework for probability is built around sets, and are useful for constructing probabilistic models in experiments.

In probability, different terminology is sometimes used. The sample space S of an experiment is the set of all possible outcomes. An event A is a subset of the sample space S.

Examples

Sets can also be equal to each other. Let A and B be two sets. If for each element x in A it is true that if x in B, then we say that A is a subset of B, written as:

More on sets

These are two facts for all sets:

1. A is always a subset of itself
2. The empty set is always a subset of A

The sets A and the empty set are called improper subsets, while any other subset is referred to as proper.

If you have taken calculus, or other mathematics courses, you may be familiar with intervals of the real number line. There are open intervals (a, b) which is a < x < b and closed intervals [a, b] a ≤ x ≤ b. As well as (a, b] and [a, b). These intervals are sets themselves and subsets of the real line. Also, (a, b) is contained in [a, b].

Sets can also be equal to each other: A = B if and only if A is contained in B and B is contained in A. For an “if and only if” statement (sometimes written as “iff”), both sides must be true. Another way to say this is that two sets are equal if and only if they have the same elements (Halmos).

Venn Diagrams

You may have seen a Venn diagram before. Consider each section, or circle in this case, of the Venn diagram as a set.

The above Venn diagram represents the union of set A and set B. In other words, it is all the points in A or B or both.

The above diagram represents the intersection of A and B. This describes all of the points in both A and B, which is the shaded region where the two circles intersect.

Now, if these two sets have no points in common, then A and B are disjoint, or mutually exclusive. The below Venn diagram represents two sets A and B that are mutually exclusive, and S is the set of all possible points:

If A is a subset of S, then the complement of A, denoted as Ā or A_C. The complement of a set are the points that are in S but not in A.

Conclusion

This post is not meant as an in depth review of set theory, but these last few equalities known as the distributive laws, are important in set theory:

Recommended reading

• Naïve Set Theory by Halmos

References:

• HALMOS, P. A. U. L. R. (2018). Naive set theory. BLURB.
• Mendelson, B. (1990). Introduction to topology. Dover.
• Blitzstein, J. K., & Hwang, J. (2015). Introduction to probability. CRC Press.