All sets to this point in the lesson have had a countable number of elements; in other words, the list of elements in each set was finite. This is called a finite set . If the elements of a finite set are listed one after another, eventually there will be no more elements to list. The number of elements in a finite set A is denoted by n ( A ) . To determine n ( A ), you can count the elements in set A .

The following are examples of finite sets and the number of elements in each set.

• A = {0, 2, 4, 6, 8, ..., 20}, n ( A ) = 11

• C ={ x | x is an integer, 1˂ x ˂ 10},  n( C ) = 8

• If T is the set of positive integers less than 12, n ( T ) = 11

• If Q is the set of letters in the word HELLO, then Q = { H , E , L , O }, n ( Q ) = 4 (Recall that elements in a set are not repeated, so L is counted only once.) An infinite set is a set that does not have a finite number of elements. You cannot list explicitly all the elements of an infinite set, so you cannot determine the number of elements in an infinite set .

The following are examples of infinite sets.

• S = {3, 6, 9,...}

• N is the set of natural numbers.

• K is the set of fractions.

There are many uses for infinite sets. One specific example comes from computer science. An important task in computer science is verifying that programs do what they are supposed to do. When programs involve loops, you need a method for showing that the loop terminates and that the program will not run forever. If your sequence of instructions creates an infinite loop, the program continues to cycle endlessly, and it can cause an entire computer system to shut down. The process of showing that a loop terminates becomes very complicated when the program is more than just a simple combination of loops. Set theory, using infinite sets, helps computer scientists sort things out by showing all the possible ways that a program can terminate. Read pages 9-11 Example 2 and pages 12-13 Example 4, and page 14 In Summary in your textbook, Principles of Mathematics 12 . Complete the Your Turn questions on page 11 and on page 13 (a, b, and c) for more practice in defining the elements in a set. Click here to verify your answers .