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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)

## Cardinality

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}