Cardinality

Learning Outcomes

  • Solve real-life problems involving sets, subsets, and cardinality properties

Often times we are interested in the number of items in a set or subset. This is called the cardinality of the set.

about the notation

In the definition of cardinality below, note that the symbol [latex]{\lvert}A{\rvert}[/latex] looks like absolute value of [latex]A[/latex] but does not denote absolute value. This symbol would be understood to represent the cardinality of set [latex]A[/latex] rather than absolute value by the context in which it is used. Note that the symbol n[latex]\left(A\right)[/latex] is also used to represent the cardinality of set [latex]A[/latex].

Cardinality

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

The cardinality of the set A is often notated as [latex]{\lvert}A{\rvert}[/latex] or n[latex]\left(A\right)[/latex]

Exercises

Let A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}.

What is the cardinality of B? A ⋃ B, A ⋂ B?

Try It

 

Exercises

What is the cardinality of P = the set of English names for the months of the year?

 

Exercises

A survey asks 200 people “What beverage do you drink in the morning”, and offers choices:

  • Tea only
  • Coffee only
  • Both coffee and tea

Suppose 20 report tea only, 80 report coffee only, 40 report both.   How many people drink tea in the morning? How many people drink neither tea or coffee?

Try It

Example

A survey asks:   Which online services have you used in the last month:

  • Twitter
  • Facebook
  • Have used both

The results show 40% of those surveyed have used Twitter, 70% have used Facebook, and 20% have used both. How many people have used neither Twitter or Facebook?

a note about the cardinality properties

The properties of cardinality can be learned by speaking them aloud, writing them out repeatedly, using flashcards, and doing practice problems with them.

Note below that the first property, spoken aloud, may be expressed as the cardinality of set A union with set B will consists of the cardinality of A together with the cardinality of B, after deducting the cardinality of their intersection. 

The second property below can be stated as the cardinality of the complement of A will consist of the cardinality of the universal set less the cardinality of A. In other words, it’s the cardinality of all the elements that are not in A.

Remember to employ more than one study strategy along with repetition and practice to learn unfamiliar mathematical concepts.

The previous example illustrated two important properties called cardinality properties:

Cardinality properties

    1. n(A ⋃ B) = n(A) + n(B) – n(A ⋂ B)
    2. n(Ac) = n(U) – n(A)

Notice that the first property can also be written in an equivalent form by solving for the cardinality of the intersection:

n(A ⋂ B) = n(A) + n(B) – n(A ⋃ B)

How was that done?

In the demonstration above, the first cardinality property was rewritten by using the property of equality as you know it from solving equations.

n(A ⋃ B) = n(A) + n(B) – n(A ⋂ B)

n(A ⋃ B) + n(A ⋂ B) = n(A) + n(B)

n(A ⋂ B) = n(A) + n(B) – n(A ⋃ B)

 

Example

Fifty students were surveyed, and asked if they were taking a social science (SS), humanities (HM) or a natural science (NS) course the next quarter.

21 were taking a SS course                  26 were taking a HM course

19 were taking a NS course                 9 were taking SS and HM

7 were taking SS and NS                     10 were taking HM and NS

3 were taking all three                         7 were taking none

How many students are only taking a SS course?

Try It

One hundred fifty people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot.

43 believed in UFOs                           44 believed in ghosts

25 believed in Bigfoot                         10 believed in UFOs and ghosts

8 believed in ghosts and Bigfoot         5 believed in UFOs and Bigfoot

2 believed in all three

How many people surveyed believed in at least one of these things?