1. 2 Chapter Numeration Systems and Sets
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2-2 Describing Sets
In the years from 1871 through 1884, Georg Cantor created set theory which has a profound effect on research and mathematical teaching. Sets, and relations between sets, are a basis for teaching children the concept of a whole number and the concept of “less than” as well as addition, subtraction, and multiplication of whole numbers.
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The Language of Sets
A set is any group or collection of objects.
The objects that belong to a set are the elements, or members, of the set.
One method of denoting a set is to simply list the elements inside braces and label the set with a capital letter.
A = {1, 2, 3, 4, 5}
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The Language of Sets
Each element of a set is listed only once.
The order of the elements is unimportant.
We symbolize that an element belongs to a set using the symbol , and we use to indicate that an element does not belong to a set.
A set must be well-defined. We must be able to tell whether or not an object belongs to the set.
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5. Listing Method (Roster Method)
Methods of Describing Sets
Listing Method (Roster Method)
C = {red, green, yellow, blue}
N = {1, 2, 3, 4, … } natural numbers
L = {a, b, c, d, …} lower case letters
Ellipsis – continues in the same manner
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6. Methods of Describing Sets
Set-Builder Notation
x is an element of the set of natural numbers
and x is less than 11
the set
of all elements x
such that
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Example 2-6
Write the following sets using set-builder notation.
a. {2, 4, 6, 8, 10, …}
b. {1, 3, 5, 7, …}
a. {x | x is an even natural number} or {x | x = 2n, n  N}
b. {x | x is an odd natural number} or {x | x = 2n − 1, n  N}
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Example 2-7a
Write the set A = {2k + 1 | k = 3, 4, 5} by listing its elements.
A = {7, 9, 11}
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Example 2-7b
Write the set B = {x|x is a positive even natural number less than 8}
B = {2, 4, 6}
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Definition
Two sets are equal if, and only if, they contain the same elements.
The order of the elements makes no difference.
If set A is equal to set B, we write A = B.
If A = B, then every element of set A is contained in set B, and every element of set B is contained in set A.
If A ≠ B , then there is at least one element that is not contained in both sets A and B.
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Definition
One-to-One Correspondence
If the elements of sets P and S can be paired so that for each element of P there is exactly one element of S and for each element of S there is exactly one element of P, then the two sets P and S are said to be in a one-to-one correspondence.
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12. One-to-One Correspondence
Set {1, 2, 3} can be placed in a one-to-one correspondence with set {a, b, c} by establishing either of the following pairings:
These are not the only possible pairs to establish a one-to-one correspondence. We would like to list all the possibilities.
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13. One-to-One Correspondence
One way to illustrate all possible one-to-one correspondences is to use a table; another is a tree. 1 2 3 a b c
b c
c b a
a c
c a b
a b
b a c 6 6
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How can we determine the number of one-to-one correspondences between two sets?
In our example, given the element 1 in the first set, there are three possible elements in the second set which can be paired with the 1, namely a, b, or c.
Once we have paired an element with 1, that leaves two elements in the second set to pair with the number 2.
Once we have paired an element in the second set with 1 and with 2, there will be one element left in the second set to pair with 3.
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Fundamental Counting Principle
If event M can occur in m ways and, after it has occurred, event N can occur in n ways, then event M followed by event N can occur in m • n ways.
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Equivalent Sets
Two sets A and B are equivalent, written A ~ B, if there exists a one-to-one correspondence between the sets.
Recall if two sets are equal, they have the same elements.
Equal sets are equivalent, but equivalent sets are not necessarily equal.
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Example 2-8
Given sets
A = {p, q, r, s} B = {a, b, c} C = {x, y, z} D = {b, a, c}
determine whether the following statements are true or false. T F T F F F F T
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Definition
The cardinal number of a set S, denoted n(S), indicates the number of elements in the set S.
What is the cardinality of each of the following?
K = {3, 5, 8, 0, 11}
L = {0}
X = {x | x is a real number and x = x + 1}
Y = Ø
n(K) = 5
n(L) = 1
n(X) = 0
n(Y) = 0
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19. Finite Sets vs. Infinite Sets
A set is finite if its cardinality is zero or a natural number.
Examples
The set of presidents of the United States.
The set of planets in our solar system.
An infinite set is a set that is not finite.
Examples
The set of whole numbers = { 0, 1, 2, 3, …}.
The set of solutions to the equation x + 1 = x + 1.
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More About Sets
The universal set U is defined as that set consisting of all elements under consideration.
We usually denote the universal set with a capital U.
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21. Universal Set Suppose the universal set is defined as:
U = {x | x is a member of the U.S. Senate}
Denote the universal set with a large rectangle, and particular sets are indicated by geometric figures inside the rectangle. In this case, only members of the U.S. Senate are included in the rectangle. This pictorial representation is a Venn diagram.
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Universal Set U
Venn Diagram
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23. Universal Set
Consider the members of the universal set, U, who are females. Since this set is wholly contained in U but does not contain all the members of U, we denote this set with a circle as follows:
F = {x | x is a female U.S. senator} U F
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24. Complement of a Set
Consider the members of the universal set, U, who are not females. This is the set of all members of the set, U, that are not in set F. We refer to this set as the complement of set F and denote it by the shaded region in the figure below.
F = {x | x is a female U.S. senator} U F
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Definition
The complement of a set F, written is the set of all elements in the universal set U that are not in F; that is,
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Example 2-9
a. If U = {a, b, c, d} and B = {c, d}, find
b. If U = {x | x is an animal in the zoo} and S = {x | x is a snake in the zoo}, describe
= {x | x is a zoo animal that is not a snake}
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Example (continued)
c. If U = N, E = {2, 4, 6, 8, …}, and O = {1, 3, 5, 7,…}, find
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Definition
Set B is a subset of set A, written if and only if every element of B is an element of A.
When a set A is not a subset of another set B, we write
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29. Subset vs. Equality of Sets
The definition allows for B to be equal to A.
Underscore
B is a subset of A
if and only if
every element of B is an element of A.
This implies this
This implies this
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Proper Subsets
Suppose set B is a subset of set A, and B is not equal to A. U B A
Then there must be at least one element of A that is not in B.
We say that B is a proper subset of A, written
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Example 2-10
Given A = {1, 2, 3, 4, 5}, B = {1, 3}, P = {x | x = 2n − 1, where n N}.
a. Which sets are subsets of each other?
Because 21 − 1 = 1, 22 − 1 = 3, 23 − 1 = 7, 24 − 1 = 15, and so on, P = {1, 3, 7, 15, …}. So,
Also
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Example (continued)
Given A = {1, 2, 3, 4, 5}, B = {1, 3}, P = {x | x = 2n − 1, where n N}.
b. Which sets are proper subsets of each other?
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33. Inequalities: An Application of Set Concepts
Suppose A and B are finite sets.
If A = B, then n(A) = n(B), and A and B are equivalent. It is possible to establish a one-to-one correspondence between A and B.
If A is a proper subset of B, then n(A) < n(B), and it is not possible to establish a one-to-one correspondence between A and B.
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Suppose A and B are infinite sets.
Let N = {1, 2, 3, 4, 5, …, n, …}
and
W = {0, 1, 2, 3, 4, …, n – 1, n, …}.
Clearly, N  W.
But it is still possible to establish a one-to-one correspondence between the sets by letting each element in N correspond to a number in W that is one smaller.
Thus, N ~ W.
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35. Number of Subsets of a Finite Set
1. If P = {a}, then P has two subsets, Ø and {a}.
2. If Q = {a, b}, then Q has four subsets, Ø, {a}, {b}, and {a, b}.
3. If R = {a, b, c}, then R has eight subsets, Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} and {a, b, c}.
In general, if there are n elements in a set, then 2n subsets can be formed.
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