Cardinality & Types of Subsets (Infinite, Finite, Equal, Empty)

Before we be begin to talk about cardinality and types of subsets, let’s review sets.

A set is a collection of elements.

An element is a collection of anything - numbers, letters, words, or objects.

Using math symbols, this means element:

The number of elements in a set is called the cardinality of the set.

The cardinality of set V = {car, truck, van, semi} is four.

There are four elements in set V .

There are two ways I have seen the symbol for cardinality.

The first has straight bars, like the absolute value symbol.

The cardinality of set V is 4.

The second way I’ve seen it written is with an n and then the set in parenthesis.

In symbols, n ( V ) = 4.

The cardinality of set V is 4.

An empty set is one that is, well, empty.

It doesn’t have any elements.

Let’s say set E is an empty set.

We can write set E in symbols like this:

The cardinality of set E is 0.

Be warned, zero is not an element in the set; it simply means the set has no elements!

Let’s look at the set of primary colors: P = {red, yellow, blue}.

We can say that set P is a finite set because it has a finite number of elements.

Finite means we can count the number of elements.

In this case, set P has 3 elements: red, yellow, and blue.

Set P has a cardinality of 3 because there are 3 elements in the set.

An infinite set is a set with an infinite number of elements.

There are two types of infinite sets - countable and uncountable.

A countable infinite set is one that can be counted in one sitting, though you may never get to the last number.

An example of a countable infinite set is the set of all integers.

An uncountable infinite set is one that cannot be counted because it is too large.

An example of an uncountable infinite set is the set of all real numbers.

The set of all real numbers equals all rational numbers and irrational numbers.

Equal sets are those that have the exact same elements in both.

Let’s say set A = {red, blue, orange} and set B = {orange, red, blue}.

Then, set A = set B .

We can say set A = set B because they have the same elements - red, blue, orange - even though they are not in order.

Equivalent sets are those that have the same cardinality, or number of elements.

Let’s say set Q is {red, blue, orange} and set R is {3, 4, 6}.

We can say set Q is equivalent to set R because they both have a cardinality of 3.

They both have 3 elements.

Equivalent doesn’t mean they have to be the same elements.

Cardinality of a set is the number of elements in that set.

It can be written like this:

An empty set is one that doesn’t have any elements. An empty set can be written like this:

A finite set has a countable finite number of elements.

An infinite set is a set with an infinite number of elements.

There are two types of infinite sets: countable and uncountable.

Equal Sets are those that have the exact same elements in both.

Equivalent Sets are those that have the same cardinality.