Cardinality

Learning Outcomes

  • Describe memberships of sets, including the empty set, using proper notation, and decide whether given items are members and determine the cardinality of a given set.
  • Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation.
  • Perform the operations of union, intersection, complement, and difference on sets using proper notation.
  • Be able to draw and interpret Venn diagrams of set relations and operations and use Venn diagrams to solve problems.
  • Recognize when set theory is applicable to real-life situations, solve real-life problems, and communicate real-life problems and solutions to others.

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

Cardinality

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

The cardinality of the set A is often notated as |A| or n(A)

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?

Example

Now, to find how many people have not used either service, we’re looking for the cardinality of (F ⋃ T)c .

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)

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?