Properties of Sets

In this lesson, we will be looking at the following topics:

• Equal Sets
• Equivalent Sets
• Order (Cardinality) of a Set
• Finite vs Infinite Sets

Equal Sets

This is really easy and simple. Two sets are said to be equal if they have the exact same elements. Not only they need to have the same number of elements, but those elements also have to be the same. The order of elements in a set does not matter, nor do duplicates.

In the above examples, $A$, $B$, and $C$ are all equal sets because they contain the same 3 elements $1, 2, 3$. Similarly, $E$ and $F$ are equal because both contain $a, b, c$. However, $A$ and $D$ are not equal because $D$ has $4$ instead of $3$.

These two sets are also equal because $B$ simplifies to just $\{1\}$ because sets do not consider duplicate values.

Equivalent Sets

Two sets are said to be equivalent if they have the same number of elements, regardless of what those elements actually are. The elements themselves can be completely different, but the sets are equivalent if their order (cardinality) is equal.

In the above examples, $P$, $Q$, and $R$ are all equivalent sets because each contains exactly 3 elements, even though those elements are different. However, $P$and $S$ are not equivalent because $S$ has 4 elements while $P$ has 3.

One to One Correspondence

Equivalency is also understood as the idea that we can establish a 1-1 correspondence between the elements of the two sets. This will play a role when discussing equivalency of infinite sets.

Cardinality of a Set

The cardinality of a set is the number of distinct elements it contains. It is denoted by placing the set’s name between two vertical bars. Repeated elements are only counted once because sets automatically remove duplicates.

The cardinality of $M$ is written as $|M| = 4$ because it contains 4 distinct elements. The cardinality of $N$ is $|N| = 5$ since there are 5 distinct vowels. The set $O$ has repeated elements, but its cardinality is $|O| = 3$ because it only contains the distinct elements $1, 2, 3$.

Singleton Set

As the same suggests, sets with the cardinality of 1, meaning sets with one and only one elements are called singleton sets. The following sets are singleton:

• $\{2\}$
• $\{3,3,3,3\}$
• $\{0\}$
• The set of all even integers between 3 and 5.
• The set of all natural numbers less than 1.
• Suns in our solar system.
• The solution set of all first degree equations of the form $ax + b = 0$

Empty Set

There is one set with the cardinality of zero. It is empty. It has no elements. We also call empty sets null sets. But it is still mathematically useful. An empty set can be denoted by these options:

• $\{ \}$
• $\emptyset$
• $\varnothing$

Some empty sets are:

• All even numbers between 2 and 4.
• The real number solution set of the equation $x^2 + 1 = 0$
• All odd numbers between 1 and 3.
• Solar system planets smaller than mercury.

Note that $\{0\}$ is not an empty or null set, it is a singleton set. $0$ is a valid element is counted as one for cardinality.

Finite and Infinite Sets

A set is called finite if it has a specific, countable number of elements. A set is called infinite if it has no end — its elements go on forever. Infinite sets can be written using patterns and ellipses ($\dots$) to show that they continue without bound.

In the above examples, $F$ and $G$ are finite sets because they contain a fixed number of elements ($|F| = 5$ and $|G| = 4$). On the other hand, $H$ is an infinite set of natural numbers, and $I$ is an infinite set of integers — neither has a last element.

Equivalency of Infinite Sets

Just like finite sets, two infinite sets are said to be equivalent if they have the same number of elements — even if those elements are different. Of course, for infinite sets, “number of elements” doesn't mean we can count them one by one. Instead, two infinite sets are equivalent if their elements can be paired one-to-one without leaving anything unmatched.

The set of natural numbers $\mathbb{N}$and the set of odd natural numbers $O$ are equivalent because each element of one set can be matched to exactly one element of the other (e.g., pair 1 with 1, 2 with 3, 3 with 5, and so on). They are not equal because their elements are not the same.

Similarly, the set of natural numbers $\mathbb{N}$ and the set of whole numbers $\mathbb{W}$ are equivalent, even though $\mathbb{W}$ includes 0 and $\mathbb{N}$ does not.

We can also see that we can make a one to one correspondence between Natural number and Integers and so they are also equivalent.

Cardinality of Infinite Sets

The cardinality of an infinite set refers to its “size” in terms of how many elements it has. Remarkably, some infinite sets that seem smaller — like the odd numbers — have the same cardinality as the entire set of natural numbers. These are called countably infinite sets, and they all share the same cardinality, often denoted by $\aleph_0$ (aleph-null).

The following sets are all countably infinite:

• Natural numbers: $\mathbb{N} = \{1, 2, 3, \dots\}$
• Whole numbers: $\{0, 1, 2, 3, \dots\}$
• Integers: $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$
• Rational numbers: $\mathbb{Q} = \left\{\frac{p}{q} \;\middle|\; p, q \in \mathbb{Z},\ q \neq 0 \right\}$

And all these countably infinite sets have the same cardinality and they are all equivalent.

The following sets are all uncountably infinite:

• Real numbers: $\mathbb{R}$
• Irrational numbers: $\mathbb{R} \setminus \mathbb{Q}$
• Points on a line segment $[0, 1]$
• All real numbers between any two distinct real numbers

But the cardinality of uncountably infinite sets is a more advanced topic to be tackled in a different course.

Activty to Identify Finite and Infinite Sets

Activity to Identify Equal and Equivalent Sets

Activity to Pair sets with the same Cardinality

Activity: Match The Columns!

Reorder the Sets B to match them with the correct Sets A.