Sets and Notation (HSC SSCE Mathematics Advanced): Revision Notes

Sets and Notation

What is a set?

A set is a collection of distinct objects or items, called elements, that are grouped together based on a shared property or characteristic. Sets provide a fundamental way to organize and work with groups of mathematical objects.

We represent sets using curly brackets (also called braces), with the elements listed inside and separated by commas. For example, the set of even numbers less than 10 can be written as:

$\{2, 4, 6, 8\}$

Another example:

$A = \{1, 2, 3, 4\}$

This represents set $A$, which contains the positive integers less than 5.

An element is an individual member or object within a set. For example, 3 is an element of the set of natural numbers $N = \{0, 1, 2, 3, 4, ...\}$.

Cardinality

The cardinality of a finite set tells us how many elements the set contains. We denote the cardinality of set $A$ using either of these notations:

$n(A) \text{ or } |A|$

Both $n(A)$ and $|A|$ represent the total number of elements in set $A$.

For example, if $A = \{1, 2, 3\}$, then:

$n(A) = 3$

This is because set $A$ contains exactly three elements: 1, 2, and 3.

The empty set

The empty set is a special set that contains no elements at all. We represent the empty set using the symbol $\varnothing$ or with empty curly brackets $\{ \}$.

$\varnothing = \{ \}$

The empty set has a cardinality of zero because it contains no elements:

$n(\varnothing) = 0$

An example of an empty set would be "the set of odd numbers divisible by 2". Since no odd number can be divisible by 2, this set contains no elements and is therefore empty.

Checking if a set is empty

To determine whether a set is empty, we need to check if it contains any elements.

Using our earlier example, is $D = \{\text{cat}, \text{dog}, \text{bird}\}$ the empty set?

No, because $D$ contains 3 elements. Since the set has elements, it is not empty.

Subsets

A set $B$ is called a subset of set $A$ (written as $B \subseteq A$) when every single element in $B$ is also an element of $A$. The notation means "all elements of $B$ are in $A$".

$A \subseteq B$

For example, if $B = \{1, 2, 3, 4\}$ and $A = \{1, 3\}$, then $A \subseteq B$ because both 1 and 3 (all elements of $A$) can be found in $B$.

Proper subsets

When set $B$ is a subset of set $A$, but $A$ contains more elements than $B$ (meaning $A$ has at least one element not in $B$), we call $B$ a proper subset of $A$. This is written as:

$B \subset A$

The key difference is that a proper subset must have strictly fewer elements than the set it is contained in.

Visual representation of subsets

Venn diagrams help us visualize subset relationships. In a Venn diagram, we draw circles to represent sets. When one set is a subset of another, we draw it as a smaller circle completely inside the larger circle.

The diagram above shows two different representations of $A \subset B$. In both cases, set $A$ (the blue inner circle) is completely contained within set $B$ (the light blue/green outer circle), illustrating the subset relationship.

Complements

The complement of a set $A$ consists of all elements that are in the universal set but are not in set $A$. The universal set, denoted by $U$, represents the complete collection of all elements under consideration for a particular problem.

We can write the complement of set $A$ in several ways:

• $\overline{A}$
• $A'$
• $A^c$

All three notations mean the same thing: the set of all elements in $U$ that are not in $A$.

Understanding the universal set

The universal set $U$ contains all possible elements relevant to the problem we're working on. Every set we consider is contained within this universal set.

For example, if $U = \{1, 2, 3, 4, 5\}$ and $A = \{1, 2\}$, then the complement $\overline{A}$ contains all elements in $U$ that are not in $A$:

$\overline{A} = \{3, 4, 5\}$