Cardinality

Cardinality of a Finite Set

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The cardinality (cardinal number) of a finite set is:

The largest element in the set

The number of elements in the set

The sum of all elements in the set

The number of subsets the set has

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Comparing Sets Without Counting

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What mathematical concept allows us to compare the sizes of two sets without explicitly counting their elements?

Subset relation

One-to-one correspondence (bijection)

Union of sets

Power set

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Definition of Equivalent Sets

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Two sets \(A\) and \(B\) are said to be equivalent (written \(A \sim B\)) if:

A and B contain exactly the same elements

A is a subset of B

A one-to-one onto correspondence can be established between A and B

A and B are both finite sets with the same sum of elements

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N is Equivalent to M = {1/n | n ∈ N}

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The table below pairs each natural number \(n\) with the fraction \(1/n\).

Participation Type	Description	Minimum Cardinality
Total Participation	Every entity in the set must be involved in at least one relationship.	1
Partial Participation	Entities in the set may or may not be involved in a relationship.	0

What can be concluded from this pairing?

N has more elements than M since N contains all positive integers

M has more elements than N since fractions are denser

N and M are equivalent sets — they have the same cardinality

N and M are equal sets — they contain the same elements

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N is Equivalent to the Set of Even Natural Numbers

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The table shows a correspondence between all natural numbers and all even natural numbers.

Term	Definition	Example
Cardinality	The total number of rows (tuples) in the relation.	500
Degree	The total number of columns (attributes) in the relation.	50

The function used is \(f(x) = 2x\). What does this demonstrate?

There are twice as many natural numbers as even numbers

The even numbers form an equivalent set to all natural numbers — they have the same cardinality

The natural numbers are uncountable

Even numbers are a proper subset, so they must have fewer elements than N

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Formal Definition of a Finite Set

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A set \(S\) is called finite and said to contain \(n\) elements if:

\(S\) contains only prime numbers

\(S \sim \{1, 2, 3, \ldots, n\}\) for some \(n \in \mathbb{N}\)

\(S\) has a largest element

\(S\) is a subset of the real numbers

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Definition of an Infinite Set

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The simplest (first) definition of an infinite set is:

A set with more than 1000 elements

A set that is not finite — counting its elements never ends

A set that contains all real numbers

A set where elements cannot be listed

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Can an Infinite Set Be Equivalent to Its Proper Subset?

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True or False: An infinite set can be equivalent to one of its own proper subsets.

True

False

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Can a Finite Set Be Equivalent to Its Proper Subset?

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True or False: A finite set can be equivalent to one of its proper subsets.

True

False

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Definition of a Countable Set

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A set \(A\) is called countable if:

It has fewer than 100 elements

It is equivalent to a subset of the positive integers

It is a subset of the rational numbers

Its elements can be written without decimal points

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Is Every Finite Set Countable?

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True or False: Every finite set is countable.

True

False

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Which of These Is an Uncountable Set?

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Which of the following sets is uncountable?

The set of all natural numbers \(\mathbb{N}\)

The set of all even natural numbers \(\{2, 4, 6, 8, ...\}\)

The set of all perfect squares \(\{1, 4, 9, 16, ...\}\)

The set of all points on a line segment

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Cardinal Number of Countably Infinite Sets

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What symbol denotes the cardinality of any countably infinite set (such as \(\mathbb{N}\))?

\(c\)

\(\aleph_0\) (aleph-null)

\(\infty\)

\(\omega\)

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Cardinal Number of the Real Numbers

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What symbol denotes the cardinality of the set of all real numbers \(\mathbb{R}\)?

\(\aleph_0\)

\(\aleph_1\)

\(c\)

\(2^{\aleph_0} + 1\)

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Which Sets Are Countably Infinite? (Multiple Answers)

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Which of the following sets are countably infinite? (Select all that apply.)

\(\{2, 4, 6, 8, \ldots\}\) — the even natural numbers

\(\{1, 4, 9, 16, \ldots\}\) — the perfect squares

\(\{1, 8, 27, 64, \ldots\}\) — the perfect cubes

The set of all points on a circle

\(\mathbb{Z}\) — the set of all integers

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Order the Cardinal Numbers

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Please drag and drop the options to sort them

Sort the following cardinal numbers in ascending order (smallest first).

\(c\)

\(3\)

\(\aleph_0\)

\(1\)

\(2^c\)

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Match Each Set to Its Cardinality Type

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To complete the line match

Click on an item in the first group
Click on an item in the second group
To delete a match, double click on a line

Set

\(\{a, b, c, d, e\}\)

\(\mathbb{N} = \{1, 2, 3, 4, ...\}\)

All points on the real line \(\mathbb{R}\)

All subsets of \(\mathbb{R}\)

Cardinality

Uncountable (cardinality \(2^c\))

Finite (cardinal number 5)

Uncountable (cardinality \(c\))

Countably infinite (cardinality ℵ₀)

Match each set to its correct cardinality classification.

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The Power Set Cardinal Number

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If a set \(X\) has cardinal number \(|X|\), what is the cardinal number of its power set \(P(X)\)?

\(2\vert{X}\vert\)

\(\vert{X}\vert^2\)

\(2^{\vert{X}\vert}\)

\(\vert{X}\vert + 1\)

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Is ℕ ~ ℤ? (Naturals Equivalent to Integers)

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Are the natural numbers \(\mathbb{N} = \{1,2,3,4,\ldots\}\) and the integers \(\mathbb{Z} = \{\ldots,-2,-1,0,1,2,\ldots\}\) equivalent sets?

No — Z has more elements since it includes 0 and negative numbers

Yes — a bijection f: Z → N exists, so they have the same cardinality ℵ₀

No — N is countable but Z is uncountable

Yes — but only because both are subsets of ℝ

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Classify These Sets by Cardinality Type

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Sets

×{1, 2, 3, ..., 50}

×{2, 4, 6, 8, ...}

×All points on the real line

×All subsets of ℕ

×ℤ (all integers)

Finite

Countably Infinite (ℵ₀)

Uncountable (≥ c)

Drag each set into the correct cardinality category.

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At Most Countable

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A set is said to be at most countable if it is:

Finite or countably infinite

Uncountable but bounded

Equivalent to ℝ

A proper subset of ℕ

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Cardinality of a Finite Set

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Explanation

Comparing Sets Without Counting

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Definition of Equivalent Sets

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N is Equivalent to M = {1/n | n ∈ N}

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Explanation

N is Equivalent to the Set of Even Natural Numbers

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Explanation

Formal Definition of a Finite Set

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Explanation

Definition of an Infinite Set

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Can an Infinite Set Be Equivalent to Its Proper Subset?

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Explanation

Can a Finite Set Be Equivalent to Its Proper Subset?

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Explanation

Definition of a Countable Set

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Is Every Finite Set Countable?

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Which of These Is an Uncountable Set?

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Cardinal Number of Countably Infinite Sets

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Cardinal Number of the Real Numbers

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Which Sets Are Countably Infinite? (Multiple Answers)

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Order the Cardinal Numbers

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Match Each Set to Its Cardinality Type

Set

Cardinality

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Explanation

The Power Set Cardinal Number

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Is ℕ ~ ℤ? (Naturals Equivalent to Integers)

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Classify These Sets by Cardinality Type

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At Most Countable

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