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Sets Math 123 September 15 Problem solving strategy 7: Draw a diagram

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Problem solving strategy 7: Draw a diagram

- A survey was taken of 150 college freshmen. Forty of them were majoring in mathematics, 30 of them were majoring in English, 20 were majoring in science, 7 had a double major of mathematics with english, and none had a double (or triple) major with science. How many students had majors other than mathematics, English, or science?

Solution method: Venn diagrams

- Digress to handout
- Solve problem: 150 - 33 - 7- 23 - 20 = 67

M

E

7

33

0

23

0

0

20

S

Why do we study sets?

- In this particular class, we use sets to describe numbers and define operations on them.
- All of mathematics can be viewed in light of what is called Set theory. Sets are the language of mathematics, in a sense.
- Venn diagrams come up in early elementary grades.

Definitions, vocabulary, etc.

- In your groups, spend some time trying to come up with definitions, explanations, and/or examples for the following (please do not use the book). Even if you have no idea what the term means, try to make up a definition. We will not go over all the terms the book goes over. When reading Section 2.1 of the book, you can skip the parts we didn’t do in class.

Elements/members of a set

Empty set

Equal sets

Finite set

Infinite set

Subset of a set

Union of sets

Intersection of sets

Difference of sets

Ordered pair

Cartesian product of sets

Definitions, vocabulary etc.

- A set is a collection of objects (any type of objects).
- A set can be described verbally, by listing its elements, or using set builder notation:
- “the set of all numbers between 1 and 5”
- {1, 2, 3, 4, 5}
- {x | 1 x 5}
Do not worry too much about this, especially the set-builder notation.

- Objects in a set are its elements/members. For example, 3 is an element of the set of all odd numbers; British Columbia is not an element of the set of all the U.S. states. Notation: x S (x is an element of S) or xS (x is not an element of S).
- A set with no elements is an empty or null set.

- Two sets are equal if they have exactly the same elements. Example: the set of all positive even numbers less than 6 and {2, 4} are equal. Notation: A = B; A ≠ B if A and B are not equal.
- A finite set is one with… well… a finite number of elements. Example: the set of all U.S. states; the set of all factors of 142452.
- An infinite set goes on forever. Example: the set of all natural numbers; the set of all numbers divisible by 142452. Don’t worry about infinite sets too much in this class.

- A Example: the set of all positive even numbers less than 6 and {2, 4} are equal. Notation: is a subset of B if every element of A is also an element of B. notation: A B. Example: {1,3} is a subset of {1,3,5}. It is possible that A and B are equal. If they are not, then A is a proper subset of B.
- The union of two sets A and B is the set that consists of elements of A and elements of B (no repetition). Notation: A B. Example: A = {1, 3, 4}, B = {2, 4 6}. A B = {1, 2, 4, 6} (not {1, 2, 3, 4, 4, 6}).

- The intersection of Example: the set of all positive even numbers less than 6 and {2, 4} are equal. Notation: A and B is the set of all elements common to both sets A and B. Notation: A B. Example (same sets as above): AB = {4}.
- The complement of a set consists of all elements that are in the universal set (the universal set is “everything”), but not the set itself. If the universal set is the set of all college students, and if A is the set of all PLU students, then the complement of A is the set of all college students not going to PLU.

- The difference of B from A is the set of all elements of A that are not in B. Notation: A - B. Example: A ={1, 3, 5}, B = {1, 4}. A - B = {3, 5}. The elements that are in B but not in A don’t matter.
- An ordered pair of numbers a and b is the pair (a,b). In this case, the order matters. That is, the set {1, 2} is the same as the set {2,1}, but the pair (1, 2) is not the same as the pair (2,1). Think of Cartesian coordinates of a point.
- The Cartesian product of A with B is the set of all ordered pairs (a,b) where a is in A and b is in B. cartesian products will be important for defining operations.

A poll of 100 registered voters designed to find out how voters kept up with current events revealed the following facts:

65 watched the news on TV

39 read the newspaper

39 listened to radio news

20 watched TV news and read the newspaper

27 watched TV news and listened to radio news

9 read the newspaper and listened to radio news

6 watched TV news, read the newspaper and listened to radio news.

How many of the 100 people surveyed kept up with current news by some means other than the three sources listed?

How many of the 100 people read the paper but did not watch the news?

How many of the 100 people used only one of he three sources listed to keep up with current events?

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