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Set Theory

What is a set?

A set is a collection of distinct objects. The objects in a set could be anything like numbers, colours, your fingers, or countries. In set notation, the objects of a set are called the elements or members of the set.

Sets can be described in three ways:

Word Description

The set NA represents all the countries in North America

Listing or Roster Notation

NA = {Canada, USA, Mexico}

Set-builder Notation

NA = {xx is any country in North America}

Set Notation and Number Sets

The set containing no elements is called the empty or null set and the notation is \(\varnothing \) or { }.

{\(\varnothing \)} is not the notation for an empty set. Instead it means that the set has one element  \(\varnothing \) 

The symbol \(\in \) represents an element of, while \(\notin \) represents not an element of

For example, Canada \(\in \) NA means Canada is an element of the set NA

Japan \(\notin \) NA means Japan is not an element of the set NA

Natural numbers \(\mathbb{N} \) (numbers used for counting). {1, 2, 3, ...} \(\in \mathbb{N} \)

Whole numbers (all natural numbers including 0). {0, 1, 2, 3, ...}

Integers \(\mathbb{Z} \) (includes 0, all natural numbers, and negative of the natural numbers). {..., -2, -1, 0, 1, 2, ...} \(\in \mathbb{Z} \)

Rational numbers  \(\mathbb{Q} \)  (all numbers that can be expressed as a fraction  \(\frac{p}{q} \), where q \(\neq \) 0). Rational numbers include all the number sets above (integers, whole, and natural numbers).

For example, \(\frac{1}{2}, \frac{2}{3}, \frac{1}{1} \)

Irrational numbers (all real numbers that are not rational numbers).

For example, \(\pi, e, \sqrt{2} \)

Real numbers \(\mathbb{R} \) (includes all the number sets above).

Complex numbers \(\mathbb{C} \) (includes real and imaginary numbers)



The number of elements in a set is called the cardinal number, or cardinality, of the set. The symbol \(n(A) \) (read "n of A") represents the cardinal number.

For example, if NA = {Canada, USA, Mexico}, then n(NA) = 3 

There are no elements in the empty set \( \varnothing \) so the cardinal number n(\(\varnothing \)) = 0

Finite and Infinite Sets

If a set is countable or has a cardinal number, that is called a finite set. Large sets are still countable for example the set of all natural numbers \(\mathbb{N} \) and the set of all rational numbers \(\mathbb{R} \). Can you think of a way to count them?

However, the set of all real numbers \(\mathbb{R} \) are uncountable. Thus, the cardinal number cannot be found. This is an example of an infinite set.

See if you can identify finite vs infinite sets

Equal Sets

Two sets are equal if all the elements appear in both sets, regardless of order.

For example, {Canada, USA, Mexico} = {Mexico, Canada, USA}

{0, 1, 2, 3} \(\neq \) {0, 2, 3}

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