Counting Elements

Number of Elements in a Set

To denote the number of elements in a set, we use the notation n(A). This is called cardinality of a set A. Putting the lowercase n in front of the set in parentheses refers to the number of elements in the set. For example: If A = (2, 4, 7, 9) we would say n(A) = 4. If we can assign a positive integer or a zero value to n(A), then set A is finite; if we cannot, then set A is infinite.

Carefully go over Example 2 in Section 7.2 in the text to see how the formula for the number of elements in the union of two sets is derived.

$n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)$

If the sets are disjoint then the last part of the formula $n\left(A\cap B\right)$ would be zero. Many students make mistakes by forgetting this last part of the formula and end up adding the elements in the intersection twice.

Let's look at some examples and try to pull the various ideas presented above into clearer focus.

Example 1. Sets with Numbers

Let A = {3, 4, 7, 8, 9} and B = {4, 7, 12, 15} in the universal set
{1, 2, 3, …, 15}.

If we want to find $n\left(A\cap B\right)$ , we would first have to determine $A\cup B=\left\{3,4,7,8,9,12,15\right\}$ . So $n(A\cup B)$ would then be 7.

To find $n\left(A\cap B\right)$ using the formula we would have to find the intersection $A\cap B=\left\{4,7\right\}$ . Then $n\left(A\cap B\right)$ is 2.

Could we use the above formula for $n\left(A\cap B\right)$ to get the same answer? Let's try it.

$\begin{array}{ll}n\left(A\cup B\right)\hfill & =n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)\hfill \\ \hfill & =5+4-2\hfill \\ \hfill & =7\hfill \end{array}$

What about A complement? Did you get $n\overline{\left(A\right)}$ ? Remember  $\bar{A}$ ={1, 2, 5, 6, 10, 11, 12, 13, 14, 15}. So the answer should be $n\overline{\left(A\right)}$ =10.

Example 2. Sets with Letters

What about sets with letters instead of numbers? Are they done differently?

$A\text{'}=\left\{a,c,d,h\right\}\text{ and }B\text{'}=\left\{b,c,d,g\right\}$ , then what is $n\left(A\text{'}\cup B\text{'}\right)$ ? Since $A\text{'}\cup B\text{'}=\left\{a,b,c,d,g,h\right\}$ , then $n\left(A\text{'}\cup B\text{'}\right)=6$ .

Example 3. Sets with Mixed Elements.

If the universal set is {0, 1, 2, …10} and A = {0, 2, 4, 6, 8, 10} and B = {prime numbers less than 10}, find the following:

$n\left(A\cap B\right)=n\left\{2\right\}=1$ (Hint: You didn't forget what the prime numbers were, did you?)

$n(A\cup B)=n\{0,2,3,4,5,6,7,8,10\}=9$

$n\left(\bar{A}\cap B\right)=n\left\{3,5,7\right\}=3$

$n\left(\bar{A}\right)=n\left\{1,3,5,7,9\right\}=5$