Lecture 1: Sets and Venn Diagrams

Frequently, we want to gather together like objects, often for the purpose of counting them.

For example, polling data will combine people into different groups based on their responses to questions.

Analysis of these results will include how responses to different questions correspond.

For example, if one of the questions is “Do you like pizza?” we could get the following data from this class.

Yes No

This forms two sets, the set of people who like pizza, and the set of people who don't.

Often we'll draw this in a Venn Diagram, letting a box represent the universe under consideration, and a circle represent any sets in that universe.

No	Yes

In this case, there are only two possibilities, a person likes pizza or does not like pizza, so one circle is sufficient to divide the universe.

Notice that the circle remains inside the universe, since this is where the objects under consideration are. It is perfectly possible that there is a cat that likes pizza, but we will ignore everything outside our universe, which is people in this class.

Let's define some notation. To make this clearer, we're going to switch to only considering the counting numbers 1 through 10.

We can write the set of numbers divisible by three in two ways.

A = { 3, 6, 9} = {6, 3, 9} = {3, 3, 6, 9, 6, 6, 6, 6}

A = {x | x is divisible by 3}

The first way writes out all of the elements of the set, enclosing them in curlly braces.

Note that it does not matter what order the elements are written in, and that repetition is ignored.

The number of elements in a set is denoted by n(A). Sometimes, I will use |A| for the same thing, but I'm going to try to stick with the notation in the book. In this case, n(A) = 3.

The second way is more convenient for larger sets, where we just say what property the objects in the set fill. Imagine trying to write out the set of numbers divisible by three if the universe we were considering was the numbers 1 to 100!

Now we can think about individual numbers.

Is 6 in our set A? We write this as 6 (in) A

Is 4 in our set A? We write this as 4 (notin) A (see book for notation)

1 2 4 5 7 8 10	3 6	9 We could also put this in a Venn diagram like the one we saw before, where the inside of the circle represents the numbers divisible by 3 and the outside represents the numbers not divisible by 3.

In cases like these, the outside is called the complement of the set. So the complement of the set of people who like pizza is the set of people who don't like pizza and the complement of the set of numbers divisible by 3 is the set of numbers not divisibe by 3.

Suppose we'd asked another question. Do you like pepperoni on your pizza?

Everyone who like pepperoni pizza clearly likes pizza, but there may be someone who doesn't like pepperoni, so we say the set of people who like pepperoni pizza is a subset of the set of people who like pizza.

No pizza	no pepperoni	pepperoni If it is the case where there is someone who doesn't like pepperoni pizza, then this is a proper subset.

Notice that by this definition a set is a subset of itself, although it is not a proper subset.

If I ask if you like anchovies on your pizza, it is possible that the set of people who answer yes has no members. This set is seen frequently, and is known as the empty set. It has it's own notation. (see book)

The set of numbers divisible by 6 is a proper subset of the set of numbers divisible by 3.

The sets do not always fall into this arrangement. Consider two sets of people, the set of women, and the set of people with long hair.

There are men with long hair, so the set of people with long hair is not a subset of the set of women.

Also, there are women with short hair, so the set of women is not a subset of the set of people with long hair.

To illustrate this arrangement, we put two overlapping circles in the universe, one for the set of women, and one for the set of people with long hair.

women	long hair	The area of overlap is the set of women with long hair. This is known as the intersection of the sets.

Also, we could look at the set of people who are female or have long hair. This is the union of the sets.

Suppose I go and collect some data. On the street, I see 50 women and 52 people with long hair. I also see 45 long-haired women and 43 short-haired men. How many people did I see, total?

To work this out, we start filling in the numbers on the Venn diagram. The easiest numbers are the intersection (45) and the outside (43). There were 50 women, and of those 45 have long hair, so only 5 women had short hair. Similarly, there are 7 men with long hair. Adding all these numbers together, I saw 100 people.

Here's an example with 3 sets. This is Exercise 7.2.40 in your book.

9	BONDS	24	0	80	3	STOCKS	5	C.O.D.	26	12 In a survey of 150 people,

111 were found to invest in stocks

98 were found to invest in bonds

100 were found to invest in certificates of deposit.

80 were found to invest in both stocks and bonds.

83 were found to invest in both bonds and certificates of deposit.

85 were found to invest in both stocks and certificates of deposit.

9 did not invest.

How many people invested in all three?

Homework: (due Friday)

Read: 7.3,7.4

Do:	7.1 11-14, 42, 67-70,

7.2 2, 4,6,8,12,14,16,18,42,43

7.3 6, 16

Elizabeth Moseman

2005-03-29T16:52:56

Elizabeth Moseman