Set Notation and Sets of Numbers Revision Notes for VCE SSCE Mathematical Methods

Set Notation and Sets of Numbers

Introduction to sets

A set is a collection of objects. The objects contained within a set are called elements or members of the set. Set notation provides a standardised way to describe and work with collections of mathematical objects.

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Set notation is fundamental to mathematics and provides a precise language for describing collections of objects. Mastering this notation will help you communicate mathematical ideas clearly and understand advanced mathematical concepts.

Element notation

When we want to indicate that a particular object belongs to a set, we use specific notation:

If $x$ is an element of set $A$ , we write $x \in A$
This can also be read as " $x$ is a member of set $A$ " or " $x$ belongs to $A$ "
If $x$ is not an element of set $A$ , we write $x \notin A$

For example, if $A = \{1, 2, 3, 4\}$ , then $2 \in A$ but $5 \notin A$ .

Subsets

A set $B$ is called a subset of set $A$ if every element of $B$ is also an element of $A$ . We write this as $B \subseteq A$ .

This expression can also be read as " $B$ is contained in $A$ " or " $A$ contains $B$ ".

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Understanding Subsets

Let $B = \{0, 1, 2\}$ and $A = \{0, 1, 2, 3, 4\}$ . Then $B \subseteq A$ because all elements of $B$ are also in $A$ .

We can verify this by checking each element:

$0 \in B$ and $0 \in A$ ✓
$1 \in B$ and $1 \in A$ ✓
$2 \in B$ and $2 \in A$ ✓

In the Venn diagram above, we can see that $3 \in A$ , $3 \notin B$ , and $B \subseteq A$ .

Intersection of sets

The intersection of two sets $A$ and $B$ is the set of elements that are common to both sets. We denote this by $A \cap B$ .

Formally, $x \in A \cap B$ if and only if $x \in A$ and $x \in B$ .

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If sets $A$ and $B$ have no elements in common, we say they are disjoint, and write $A \cap B = \emptyset$ , where $\emptyset$ is called the empty set.

Union of sets

The union of sets $A$ and $B$ is the set of elements that are in $A$ or in $B$ (or in both). We denote this by $A \cup B$ .

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The key difference between intersection and union:

Intersection ( $\cap$ ): Elements must be in both sets
Union ( $\cup$ ): Elements must be in at least one set

For example, let $A = \{1, 3, 5, 7, 9\}$ and $B = \{1, 2, 3, 4, 5\}$ . Then:

$A \cap B = \{1, 3, 5\}$ (the common elements)
$A \cup B = \{1, 2, 3, 4, 5, 7, 9\}$ (all elements from both sets)

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Worked Example: Finding Intersection and Union

For $A = \{1, 2, 3, 7\}$ and $B = \{3, 4, 5, 6, 7\}$ , find:

a) $A \cap B$

A \cap B = \{3, 7\}

The elements $3$ and $7$ appear in both sets.

b) $A \cup B$

A \cup B = \{1, 2, 3, 4, 5, 6, 7\}

The union contains all elements that belong to $A$ or $B$ (or both).

Set difference

The set difference of two sets $A$ and $B$ is given by:

A \setminus B = \{ x : x \in A, x \notin B \}

The set $A \setminus B$ contains the elements of $A$ that are not elements of $B$ .

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Worked Example: Finding Set Differences

For $A = \{1, 2, 3, 7\}$ and $B = \{3, 4, 5, 6, 7\}$ , find:

a) $A \setminus B$

A \setminus B = \{1, 2, 3, 7\} \setminus \{3, 4, 5, 6, 7\}

= \{1, 2\}

The elements $1$ and $2$ are in $A$ but not in $B$ .

b) $B \setminus A$

B \setminus A = \{3, 4, 5, 6, 7\} \setminus \{1, 2, 3, 7\}

= \{4, 5, 6\}

The elements $4$ , $5$ , and $6$ are in $B$ but not in $A$ .

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Notice: $A \setminus B$ and $B \setminus A$ are generally not the same! Set difference is not commutative.

Sets of numbers

Mathematics uses several important sets of numbers, each with its own notation and properties.

Natural numbers

The natural numbers are the counting numbers: $\{1, 2, 3, 4, \ldots\}$

We denote the set of natural numbers by $\mathbb{N}$ .

Integers

The integers include all whole numbers, both positive and negative, as well as zero: $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$

We denote the set of integers by $\mathbb{Z}$ .

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The symbol $\mathbb{Z}$ comes from the German word "Zahlen," which means "numbers."

Rational numbers

Rational numbers are numbers that can be expressed as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ .

We denote the set of rational numbers by $\mathbb{Q}$ .

An important characteristic of rational numbers is that they can be written as either terminating decimals (like $0.5$ or $2.75$ ) or recurring decimals (like $0.\overline{3}$ or $0.142857\overline{142857}$ ).

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The symbol $\mathbb{Q}$ comes from the word "quotient," since rational numbers are quotients of integers.

Irrational numbers

Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. Examples include $\pi$ and $\sqrt{2}$ .

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Irrational numbers have decimal expansions that neither terminate nor repeat. For example, $\pi = 3.14159265...$ continues forever without any repeating pattern.

Real numbers

The real numbers include all rational and irrational numbers. We denote the set of real numbers by $\mathbb{R}$ .

The hierarchy of number sets

These number sets form a hierarchy where each set is contained within the next:

\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}

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Understanding the Hierarchy:

Every natural number is an integer ( $\mathbb{N} \subseteq \mathbb{Z}$ )
Every integer is a rational number ( $\mathbb{Z} \subseteq \mathbb{Q}$ )
Every rational number is a real number ( $\mathbb{Q} \subseteq \mathbb{R}$ )
But NOT every real number is rational (irrational numbers exist!)

This relationship can be visualised as follows:

The diagram shows that natural numbers are a subset of integers, integers are a subset of rationals, and rationals are a subset of real numbers.

Describing sets using set-builder notation

When listing all elements of a set is impractical or impossible (particularly for infinite sets), we use set-builder notation. This notation uses the format $\{x : \text{condition}\}$ , which reads as "the set of all $x$ such that condition holds".

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Set-builder notation provides a powerful and concise way to describe sets by specifying the properties their elements must satisfy, rather than listing every element.

Examples include:

$\{x : 0 < x < 1\}$ is the set of all real numbers strictly between $0$ and $1$
$\{x : x \geq 3\}$ is the set of all real numbers greater than or equal to $3$
$\{x : x > 0, x \in \mathbb{Q}\}$ is the set of all positive rational numbers
$\{2n : n = 0, 1, 2, \ldots\}$ is the set of all non-negative even numbers
$\{2n + 1 : n = 0, 1, 2, \ldots\}$ is the set of all non-negative odd numbers

Interval notation

Among the most important subsets of $\mathbb{R}$ are the intervals. Intervals represent continuous ranges of real numbers and use special notation involving brackets and parentheses.

For real numbers $a$ and $b$ where $a < b$ :

Interval notation	Set-builder notation	Description
$(a, b)$	$\{x : a < x < b\}$	Open interval (excludes both endpoints)
$[a, b]$	$\{x : a \leq x \leq b\}$	Closed interval (includes both endpoints)
$(a, b]$	$\{x : a < x \leq b\}$	Half-open interval (excludes $a$ , includes $b$ )
$[a, b)$	$\{x : a \leq x < b\}$	Half-open interval (includes $a$ , excludes $b$ )
$(a, \infty)$	$\{x : a < x\}$	Infinite interval (all numbers greater than $a$ )
$[a, \infty)$	$\{x : a \geq x\}$	Infinite interval (all numbers greater than or equal to $a$ )
$(-\infty, b)$	$\{x : x < b\}$	Infinite interval (all numbers less than $b$ )
$(-\infty, b]$	$\{x : x \leq b\}$	Infinite interval (all numbers less than or equal to $b$ )

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Bracket Notation Rules:

Round brackets $($ or $)$ exclude the endpoint (open)
Square brackets $[$ or $]$ include the endpoint (closed)
Infinity symbols ( $\infty$ and $-\infty$ ) always use round brackets because infinity is not a real number

Visual representation of intervals

Intervals can be represented on number lines. The following conventions are used:

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Number Line Conventions:

A closed circle (•) indicates that the number is included in the interval
An open circle (◦) indicates that the number is not included in the interval

Special notation for real numbers

The following subsets of real numbers have special notation:

Positive real numbers: $\mathbb{R}^+ = \{x : x > 0\}$
Negative real numbers: $\mathbb{R}^- = \{x : x < 0\}$
Real numbers excluding zero: $\mathbb{R} \setminus \{0\}$

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Key Points to Remember:

Element notation: Use $x \in A$ to show $x$ is an element of set $A$ , and $x \notin A$ to show it is not
Subset: $B \subseteq A$ means every element of $B$ is also in $A$
Intersection ( $A \cap B$ ): Contains only elements that appear in both sets
Union ( $A \cup B$ ): Contains all elements from either set (or both)
Set difference ( $A \setminus B$ ): Contains elements in $A$ but not in $B$
Number hierarchy: $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$ (natural numbers ⊂ integers ⊂ rationals ⊂ real numbers)
Interval notation: Round brackets $($ or $)$ exclude endpoints, square brackets $[$ or $]$ include endpoints
Set-builder notation: $\{x : \text{condition}\}$ describes sets by specifying properties that elements must satisfy

Set Notation and Sets of Numbers (VCE SSCE Mathematical Methods): Revision Notes