As with the section on logic it should be noted that the following information on set theory is only an introduction. Hopefully children will be interested enough to want to know more either now or in the future.
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.
Example.
The following are sets.
a. the set of all men
b. the set of whole numbers between 17
Example. The following are not well defined, so are not sets.
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.
Example.
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.
Example.
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,
Part 1. Determine the following sets.
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1. The set of odd numbers between 16 and 24
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2. The set of states whose names begin with “New.”
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3. The set of whole numbers between 2 and 20 which are exactly divisible by 3.
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4. The set of all foreigners elected President of the U.S.
Part 2. Using the sets below answer the following questions yes or no:
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?
Solutions Part 1.
1. { 17, 19, 21, 23}
2. { New York, New Jersey, New Hampshire, New Mexico)
3. { 3,6,9,12,15,18}
4. ¯
Part 2.
1. No
2. Yes
3. Yes
4. Yes
5. No
6. Yes