## 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

One can describe a set by
• 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 $$\{1,3,5,\ldots,99\}$$ to specify the set of odd integers from $$1$$ up to $$99$$, and $$\{4,8,12,\ldots\}$$ 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 $$1\leq n \leq 100\}$$ 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 $$\{ ax^2 + bx + c : a,b,c \in \mathbb{R}\}$$.

## Special sets

• $$\emptyset$$ denotes the empty set, the set with no members.
• $$\mathbb{N}$$ denotes the set of natural numbers; i.e. $$\{1,2,3,\ldots\}$$.

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

• $$\mathbb{Q}$$ 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, $$\mathbb{Z}^3 = \left \{ \begin{bmatrix} a\\ b\\ c\end{bmatrix} : a, b, c \in \mathbb{Z}\right\}$$.

## Common Set Notation

Let $$A$$ and $$B$$ be sets.

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

• $$A = B$$ 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 $$A \cup B = \{1,2,3\}$$.

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

## 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 $$A \backslash B$$.

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

1. 0

2. 5

3. $$\sqrt{3}$$