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]
Example – cARDINALITY
Let A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}.
What is the cardinality of B? A ⋃ B, A ⋂ B?
Click here to see a video showing how this example is worked.
What is the cardinality of P = the set of English names for the months of the year?
Try It
ExAMPLE – Using Cardinality and Venn diagrams in surveying
Sometimes we may be interested in the cardinality of the union or intersection of sets, but not know the actual elements of each set. This is common in surveying.
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?
This question can most easily be answered by creating a Venn diagram. We can see that we can find the people who drink tea by adding those who drink only tea to those who drink both: 60 people.
We can also see that those who drink neither are those not contained in the any of the three other groupings, so we can count those by subtracting from the cardinality of the universal set, 200.
200 – 20 – 80 – 40 = 60 people who drink neither.
Try It
Example
A survey asks: Which online services have you used in the last month:
- 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?
The previous example illustrated two important properties called cardinality properties:
Cardinality properties
-
- n(A ⋃ B) = n(A) + n(B) – n(A ⋂ B)
- n(Ac) = n(U) – n(A)
a note about the cardinality properties
Note that the first cardinal property (above), 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 cardinal property 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.
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?
