Sets And Counting
SECTION 6.1 PROBLEM SET: SETS AND COUNTING Find the indicated sets
1) List all subsets of the following set.
{ Al, Bob}
2) List all subsets of the following set.
{ Al, Bob, Chris}
3) List the elements of the following set.
{ Al, Bob, Chris, Dave}
{Bob, Chris, Dave, Ed}4) List the elements of the following set.
{ Al, Bob, Chris, Dave}
{Bob, Chris, Dave, Ed}In Problems 5 - 8, let Universal set = U = {a, b, c, d, e, f, g, h, i, j}, V = {a, e, i, f, h}, and W = { a, c, e, g, i}.
List the members of the following sets.
5) V
W 6) V
W7) VW 8) VW
In 9 - 12, let Universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, B = { 1, 3, 4, 6}, and C = {2, 4, 6}.
List the members of the following sets.
9) A
B 10) A
C11) ABC 12) ABC
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CHAPTER 6 PROBLEM SETS Name:_________________________________
Find the number of elements in the following sets.
13) In Mrs. Yamamoto's class of 35 students, 12 students are taking history, 18 are taking English, and 4 are taking both. Draw a Venn diagram and determine how many students are taking neither history nor English?
14) In the County of Santa Clara 700,000 people read the San Jose Mercury News, 400,000 people read the San Francisco Examiner, and 100,000 read both newspapers. How many read either the Mercury News or the Examiner?
15) A survey of athletes revealed that for their minor aches and pains, 30 used aspirin, 50 used ibuprofen, and 15 used both. How many athletes were
surveyed?
16) In a survey of computer users, it was found that 50 use HP printers, 30 use IBM printers, 20 use Apple printers, 13 use HP and IBM, 9 use HP and Apple, 7 use IBM and Apple, and 3 use all three. How many use at least one of these Brands?
17) This quarter, a survey of 100 students at De Anza college finds that 50 take math, 40 take English, and 30 take history. Of these 15 take English and math, 10 take English and history, 10 take math and history, and 5 take all three subjects. Draw a Venn diagram and determine the following.
a) The number of students taking math but not the other two subjects.
b) The number of students taking English or math but not history.
c) The number of students taking none of these subjects.
18) In a survey of investors it was found that 100 invested in stocks, 60 in mutual funds, and 50 in bonds. Of these, 35 invested in stocks and mutual funds, 30 in mutual funds and bonds, 28 in stocks and bonds, and 20 in all three. Determine the following.
a) The number of investors that participated in the survey.
b) How many invested in stocks or mutual funds but not in bonds?
c) How many invested in exactly one type of investment?
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6.2 Tree Diagrams and the Multiplication Axiom
In this chapter, we are trying to develop counting techniques that will be used in the next chapter to study probability. One of the most fundamental of such techniques is called the Multiplication Axiom. Before we introduce the multiplication axiom, we first look at some examples.
Example 1 If a woman has two blouses and three skirts, how many different outfits consisting of a blouse and a skirt can she wear?
Solution: Suppose we call the blouses b1 and b2, and skirts s1, s2, and s3. We can have the following six outfits.
b1s1, b1s2, b1s3, b2s1, b2s2, b2s3 Alternatively, we can draw a tree diagram:
b1
The tree diagram gives us all six possibilities. The method involves two steps. First the woman chooses a blouse. She has two choices: blouse one or blouse two. If she chooses blouse one, she has three skirts to match it with; skirt one, skirt two, or skirt three. Similarly if she chooses blouse two, she can match it with each of the three skirts, again. The tree diagram helps us visualize these possibilities.
The reader should note that the process involves two steps. For the first step of choosing a blouse, there are two choices, and for each choice of a blouse, there are three choices of choosing a skirt. So altogether there are 2
.
3 = 6 possibilities.If, in the above example, we add the shoes to the outfit, we have the following problem.
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Chapter 6: Sets And Counting 151
Example 2 If a woman has two blouses, three skirts, and two pumps, how many different outfits consisting of a blouse, a skirt, and a pair of pumps can she wear?
Solution: Suppose we call the blouses b1 and b2, the skirts s1, s2, and s3, and the pumps p1, and p2.
We count the number of branches in the tree, and see that there are 12 different possibilities.
This time the method involves three steps. First, the woman chooses a blouse. She has two choices: blouse one or blouse two. Now suppose she chooses blouse one. This takes us to step two of the process which consists of choosing a skirt. She has three choices for a skirt, and let us suppose she chooses skirt two. Now that she has chosen a blouse and a skirt, we have moved to the third step of choosing a pair of pumps. Since she has two pairs of pumps, she has two choices for the last step. Let us suppose she chooses pumps two. She has chosen the outfit consisting of blouse one, skirt two, and pumps two, or b1s2p2. By looking at the different branches on the tree, one can easily see the other possibilities.
The important thing to observe here, again, is that this is a three step process. There are two choices for the first step of choosing a blouse. For each choice of a blouse, there are three choices of choosing a skirt, and for each combination of a blouse and a skirt, there are two choices of selecting a pair of pumps. All in all, we have 2
.
3.
2 = 12 different possibilities.The tree diagrams help us visualize the different possibilities, but they are not practical when the possibilities are numerous. Besides, we are mostly interested in finding the number of elements in the set and not the actual possibilities. But once the problem is envisioned, we can solve it without a tree diagram. The two examples we just solved may have given us a clue to do just that.
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Let us now try to solve Example 2 without a tree diagram. Recall that the problem involved three steps: choosing a blouse, choosing a skirt, and choosing a pair of pumps. The number of ways of choosing each are listed below.
The number of ways
By multiplying these three numbers we get 12, which is what we got when we did the problem using a tree diagram.
The procedure we just employed is called the multiplication axiom.