Introduction

Sets are arguably the most fundamental objects in modern mathematics. Familiarity with set notation is a prerequisite to reading post-secondary mathematics. What follows is a brief summary of key definitions and concepts related to sets required in this course.

Definition

A set is a well-defined collection of distinct mathematical objects. The objects are called members or elements of the set.

Describing sets

• specifying a rule or a verbal description. For example, one can say “let $A$ be the set of all odd integers”. Then $A$ is a set and its elements are all the odd integers.

• enclosing the list of members within curly brackets. For example, $C=\{2,4,5\}$ denotes a set of three numbers: 2, 4, and 5, and $D=\{(2,4),(-1,5)\}$ denotes a set of two pairs of numbers.

  Abbreviations can be used if the set is large or infinite. For example, one may write $\{1,3,5,\dots ,99\}$ to specify the set of odd integers from $1$ up to $99$, and $\{4,8,12,\dots \}$ to specify the (infinite) set of all positive integer multiples of $4$.

  Another option is to use set-builder notation: $F=\{n^3:n$ is an integer with $1\le n\le 100\}$ is the set of cubes of the first $100$ positive integers. (The way to interpret this is as follows: $F$ is the set of all $n^3$ such that $n$ is an integer from $1$ to $100$.)

Examples

1. The set of all even integers is given by $\{2n:n\text{ is an integer }\}$.

2. The set of all polynomials in $x$ with real coefficients having degree at most two is given by $\{a{x}^2+bx+c:a,b,c\in \mathbb{R}\}$.

Special sets

• $\mathrm{\varnothing }$ denotes the empty set, the set with no members.

• $\mathbb{N}$ denotes the set of natural numbers; i.e. $\{1,2,3,\dots \}$.

• $\mathbb{Z}$ denotes the set of integers; i.e. $\{\dots ,-2,-1,0,1,2,\dots \}$.

• $\mathbb{Q}$ denotes the set of rational numbers (the set of all possible fractions, including the integers).

• $\mathbb{R}$ denotes the set of real numbers.

• $\mathbb{C}$ denotes the set of complex numbers. (This set will be introduced more formally later.)

Sets of $n$-tuples

There is a convenient notation for specifying sets of $n$-tuples whose entries are from the same set.

Let $A$ be a set. Let $n$ be a positive integer. Then, the set of $n$-tuples whose entries are elements of $A$ is denoted by $A^n$.

For example, ${\mathbb{Z}}^3$ is the set of all $3$-tuples whose entries are integers. In other words, ${\mathbb{Z}}^3=\{\left[\begin{array}{c}a\\ b\\ c\end{array}\right]:a,b,c\in \mathbb{Z}\}$.

Common Set Notation

Let $A$ and $B$ be sets.

• $|A|$, called cardinality of $A$, denotes the number of elements of $A$. For example, if $A=\{(1,2),(3,4)\}$, then $|A|=2$.

• $A=B$ if and only if they have precisely the same elements. For example, if $A=\{4,9\}$ and $B=\{n^2:n=2$ or $n=3\}$, then $A=B$.

• $A\subseteq B$ if and only if every element of $A$ is also an element of $B$. We call $A$ a subset of $B$. For example, $\{1,8,1107\}\subseteq \mathbb{N}$.

• $a\in A$ means $a$ is a member of $A$. For example, $5\in \mathbb{Q}$

• $a\notin A$ means $a$ is not a member of $A$. For example, $\frac{2}{7}\notin \mathbb{Z}$

• $A\cap B$ denotes the set containing elements that are in both $A$ and $B$. $A\cap B$ is called the intersection of $A$ and $B$. For example, if $A=\{1,2\}$ and $B=\{2,3\}$, then $A\cap B=\{2\}$.

• $A\cup B$ denotes the set containing elements that are in either $A$ or $B$ or both. $A\cup B$ is called the union of $A$ and $B$. For example, if $A=\{1,2\}$ and $B=\{2,3\}$, then $A\cup B=\{1,2,3\}$.

• $A\mathrm{\setminus }B$ denotes the set containing elements that are in $A$ but not in $B$. $A\mathrm{\setminus }B$ is read as “$A$ drop $B$”. For example, if $A=\{1,2\}$ and $B=\{2,3\}$, then $A\mathrm{\setminus }B=\{1\}$.

Quick Quiz

Exercises

1. Let $A=\{1,3,5,7\}$ and $B=\{0,3,6,7,9\}$. Write out the following sets: $A\cup B$, $A\cap B$, and $A\mathrm{\setminus }B$.  

   • $A\cup B=\{0,1,3,5,6,7,9\}$

   • $A\cap B=\{3,7\}$

   • $A\cup B=\{1,5\}$

2. Which of the following are members of $\mathbb{Q}\cap \{a:a\in \mathbb{R}$ and $a>\sqrt{2}\}$?

   1. 0  

      Not a member.

   2. 5  

      A member.

   3. $\sqrt{3}$  

      Not a member.