# Problem Solving

## CONTENTS OF CURRICULUM UNIT 80.07.04

## Logic and Set Theory

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Set Theory
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A set is a well defined collection of “objects.” The term “well defined” means that the set is described in such a way that we can determine whether or not any given object belongs to that set.

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Example.
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The following are sets.

a. the set of all men

b. the set of whole numbers between 17

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Example. The following are not well defined, so are not sets.
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a.) all wellknown artists

b.) three wealthy men

Every set is a collection of elements or members of that set. When there are only a few members of a set we usually use the tabulation or roster method to denote the set.

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Example.
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The set of whole numbers between 1 and 4 would be written { 2,3} . The elements of the set are all listed between the set braces.

The tabulation method can also be used when a set has a great or even infinite number of elements, provided that those elements are ordered.

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Example.
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The set of whole numbers less than 50 can be denote {0,1,2....,49}

The set of all whole numbers would be {0,1,2,...}.

Note: The first three elements are shown, then 3 dots and finally the last element if the set is not infinite.

Another method for naming a set is the descriptive method which is used when the above tabulation method is not convenient or will not work. The diagram shows the descriptive method:

(figure available in print form)

We also use the symbol · to denote the phrase “is an element of.” Thus we can say, “If A= {2,3,4), then A is a set containing 3 elements. 2 £ A, 3 · A, and 4 · A.”

The next ideas to be discussed are the universal and the null set. The universal set describes that set from which we select all our sets in a particular problem. The null set on the other hand is a set which contains no elements. The universal set usually is shown by U and the null set by ~ .

If U= {1,2,3,4,5} and B= {1,2} and C= {3,4} then the set of all elements common to B and C is the null set ¯ since no element of B is an element of C. All of the elements of B and C however are elements of U, the universal set.

Equal sets are sets in which both elements of set A and elements of set B are the same written A> B. If A = {1,2,3} and B= {3,1,2} they are equal sets. Any element of A is an element of B and any element of B is an element of A.

Subsets are sets in which all of the elements of set A are elements of set B yet not all of the elements of set B are elements of set A. For example A= {1,2}, B= {1,2,3}. Set A is a subset of set B, written A ² B, Note: the ¯ is always a subset of another set since there are never any elements in ¯ which are not in any other set A. Therefore ¯²A,

The following exercises relate to the material introduced to this point,

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Part 1. Determine the following sets.
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- 1. The set of odd numbers between 16 and 24
- 2. The set of states whose names begin with “New.”
- 3. The set of whole numbers between 2 and 20 which are exactly divisible by 3.
- 4. The set of all foreigners elected President of the U.S.

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Part 2. Using the sets below answer the following questions yes or no:
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U = { 1,2,3,4,5,6}

A = { 1,2,5}

B = { 1,2,3,4,5}

C = { 1,5,2}

D ={6}

1. Is A = D?

2. Is D ² U?

3. Is C = A?

4. Is A ² B?

5. Is ¯ = B?

6. Is ¯ ² B?

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Solutions Part 1.
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1. { 17, 19, 21, 23}

2. { New York, New Jersey, New Hampshire, New Mexico)

3. { 3,6,9,12,15,18}

4. ¯

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Part 2.
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1. No

2. Yes

3. Yes

4. Yes

5. No

6. Yes