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.
Notation: The cardinality of the set [latex]A[/latex] is often notated as [latex]{\lvert}A{\rvert}[/latex] or [latex]n\left(A\right)[/latex]
In the definition of cardinality above, note that the symbol [latex]{\lvert}A{\rvert}[/latex] looks like the absolute value of [latex]A[/latex] but does not denote the 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 [latex]n\left(A\right)[/latex] is also used to represent the cardinality of set [latex]A[/latex].Let [latex]A = \{1, 2, 3, 4, 5, 6\}[/latex] and [latex]B = \{2, 4, 6, 8\}[/latex]. What is the cardinality of:
[latex]B[/latex]
[latex]A \cup B[/latex]
[latex]A \cap B[/latex]
The cardinality of [latex]B[/latex] is [latex]4[/latex], since there are [latex]4[/latex] elements in the set.
The cardinality of [latex]A \cup B[/latex] is [latex]7[/latex], since [latex]A \cup B = \{1, 2, 3, 4, 5, 6, 8\}[/latex], which contains [latex]7[/latex] elements.
The cardinality of [latex]A \cap B[/latex] is [latex]3[/latex], since [latex]A \cap B = \{2, 4, 6\}[/latex], which contains [latex]3[/latex] elements.
It is possible to identify the cardinality of unions, intersections, and complements of sets. In order to do so we must apply the cardinality properties.
A note about the cardinality properties
You’ve already seen how to use the properties of real numbers and how they can be written as “templates” or “forms” in the general case. The properties of cardinality, although they are not the same as number properties, can be learned in a similar way, by speaking them aloud, writing them out repeatedly, using flashcards, and doing practice problems with them.
Remember to employ more than one study strategy along with repetition and practice to learn unfamiliar mathematical concepts.
Fifty students were surveyed and asked if they were taking a social science (SS), humanities (HM), or natural science (NS) course the next quarter.
[latex]21[/latex] were taking a SS course
[latex]19[/latex] were taking a NS course
[latex]7[/latex] were taking SS and NS
[latex]3[/latex] were taking all three
[latex]26[/latex] were taking a HM course
[latex]9[/latex] were taking SS and HM
[latex]10[/latex] were taking HM and NS
[latex]7[/latex] were taking none
How many students are only taking a SS course?
It might help to look at a Venn diagram.
Let’s map the survey results to the regions of the Venn diagram:
[latex]21[/latex] students in regions [latex]a, b, d[/latex], and [latex]e[/latex]
[latex]19[/latex] students in regions [latex]g, d, f[/latex], and [latex]e[/latex]
[latex]7[/latex] students in regions [latex]d[/latex], and [latex]e[/latex]
[latex]3[/latex] students in region [latex]e[/latex]
[latex]26[/latex] students in regions [latex]c, b, f[/latex], and [latex]e[/latex]
[latex]9[/latex] students in regions [latex]b[/latex], and [latex]e[/latex]
[latex]10[/latex] students in regions [latex]f[/latex], and [latex]e[/latex]
[latex]7[/latex] students in region [latex]h[/latex]
Since [latex]7[/latex] students were taking a SS and NS course, we know that [latex]n(d) + n(e) = 7[/latex]. Since we know there are [latex]3[/latex] students in region [latex]e[/latex], there must be [latex]7 – 3 = 4[/latex] students in region [latex]d[/latex].
Similarly, since there are [latex]10[/latex] students taking HM and NS, which includes regions [latex]e[/latex] and [latex]f[/latex], there must be
[latex]10 – 3 = 7[/latex] students in region [latex]f[/latex].
Since [latex]9[/latex] students were taking SS and HM, there must be [latex]9 – 3 = 6[/latex] students in region [latex]b[/latex].
Now, we know that [latex]21[/latex] students were taking a SS course. This includes students from regions [latex]a, b, d,[/latex], and [latex]e[/latex]. Since we know the number of students in all but region [latex]a[/latex], we can determine that [latex]21 – 6 – 4 – 3 = 8[/latex] students are in region [latex]a[/latex].
[latex]8[/latex] students are taking only a SS course.
