A set is a unique collection of well-defined objects.

Objects in a set are called elements or members of the set.

We often refer to sets in everyday language: you may have heard of a tea set, a drum set, a set of action figurines, etc. These terms refer to specific collections of objects. It is clear which objects in are members of each set and which are not.

Sets are commonly notated using curly braces. For example:

  • The set of numbers from 1 to 3: A={1,2,3}A = \{1, 2, 3\}
  • A set of words: B={hello,world}B = \{\text{hello}, \text{world}\}


The elements of a set must be well-defined, and there should be no ambiguity about whether something is a set member or not.

We use the \in notation to describe if something is a set member:

  • 1A1 \in A
  • helloB\text{hello} \in B

We use the \notin notation to describe if something is a not set member:

  • 99A99 \notin A
  • goodbyeB\text{goodbye} \notin B

The symbols in Latex are:

  • \in: \in
  • \notin: \notin

A set can be a member of another set. Interestingly, this fact means that the definition of a "set" is circular, making it technically an undefined term.


Sets do not have duplicates; therefore:

A={1,1,2}={1,2}A = \{1, 1, 2\} = \{1, 2\}

This property means sets are practically helpful for finding unique counts of things, and they often appear in programming for this purpose and many others.


If every element in set AA is in set BB, we consider set AA to be a subset of BB. The notation for subset is ABA \subseteq B. In LaTeX, we use: \subseteq

For example, the set of days on the weekend is a subset of the days in the week:

{Sunday,Saturday}{Sunday,Monday,Tuesday,Wednesday,Thursday,Friday,Saturday}\{\text{Sunday}, \text{Saturday}\} \subseteq \{\text{Sunday}, \text{Monday}, \text{Tuesday}, \text{Wednesday}, \text{Thursday}, \text{Friday}, \text{Saturday}\}


Conversely, we consider BB to contain AA, denoted as BAB \supseteq A. We say that BB is a superset of AA.

If AA is not a subset of BB, we write ABA \nsubseteq B. In LaTeX, it's: \nsubseteq


The number of elements in a set is called Cardinality

Equal Sets

When 2 sets contain the same elements, we consider them equal sets: A==BA == B.

Empty Set

When a set has no elements it's called the empty set: \emptyset: ={}\emptyset = \{\}. The empty set is a subset of every other set.

Universal Set

A special set called the universal set UU, is a set where every other set is a subset of UU.

AUA \subseteq U for every set AA

Disjoint Sets

A set may not have common elements. These are called disjoint sets. For example, a set of integers and a set of letters are disjoint sets.

Power Set

The power set is a subset representing all subsets of set A, written as P(A)P(A).

For example, the power set of A={1,2}A = \{1, 2\}:

P(A)={,{1},{2},{1,2}}P(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}

Infinite and finite sets

If a set has an infinite number of elements, we call it an infinite set.

Some Special Infinite Set come up frequently in Math.

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