Cardinality

Test your understanding of set sizes — Finite, Countable, and Uncountable Sets

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QUESTION OF
Views #: 4
Questions #: 21
Time: 20 minutes
Pass Score: 80.0%
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Mode

Cardinality of a Finite Set

1 pts
volume_mute

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

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Comparing Sets Without Counting

1 pts
volume_mute

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

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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:

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N is Equivalent to M = {1/n | n ∈ N}

1 pts
volume_mute

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?

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N is Equivalent to the Set of Even Natural Numbers

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volume_mute

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?

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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:

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Definition of an Infinite Set

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

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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.

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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.

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Definition of a Countable Set

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

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Is Every Finite Set Countable?

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True or False: Every finite set is countable.

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Which of These Is an Uncountable Set?

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Which of the following sets is uncountable?

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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 \(\mathbb{N}\))?

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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 \(\mathbb{R}\)?

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Which Sets Are Countably Infinite? (Multiple Answers)

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

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Order the Cardinal Numbers

2 pts
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

2 pts

Match each set to its correct cardinality classification.

To complete the line match

  1. Click on an item in the first group
  2. Click on the match 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)
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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 \(P(X)\)?

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Is ℕ ~ ℤ? (Naturals Equivalent to Integers)

1 pts
volume_mute

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?

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Classify These Sets by Cardinality Type

3 pts

Drag each set into the correct cardinality category.

drag and drop the selected option to the right place

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At Most Countable

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

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Keywords
Year 12