Chapter 6: Set Theory 6.1 Set Theory: Definitions and the Element Method of Proof 1 6.1 Set Theory -...
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Transcript of Chapter 6: Set Theory 6.1 Set Theory: Definitions and the Element Method of Proof 1 6.1 Set Theory -...
6.1 Set Theory - Definitions and the Element Method of Proof
1
Discrete Structures
Chapter 6: Set Theory
6.1 Set Theory: Definitions and the Element Method of Proof
The introduction of suitable abstractions is our only mental aid to organize and master complexity.– E. W. Dijkstra, 1930 – 2002
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6.1 Set Theory - Definitions and the Element Method of Proof
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Subsets
• Let’s write what it means for a set A to be a subset of a set B as a formal universal conditional statement:
A B x, if x A then x B.
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6.1 Set Theory - Definitions and the Element Method of Proof
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Subsets
• The negation is existential
A B x, if x A and x B.
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6.1 Set Theory - Definitions and the Element Method of Proof
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Subsets
• A proper subset of a set is a subset that is not equal to its containing set.
A is a proper subset of B 1. A B, and 2. there is at least one element
in B that is not in A.
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6.1 Set Theory - Definitions and the Element Method of Proof
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Element Argument
• Let sets X and Y be given. To prove that X Y,
1. Suppose that x is a particular but arbitrarily chosen element of X,
2. Show that x is an element of Y.
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6.1 Set Theory - Definitions and the Element Method of Proof
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Example – pg. 350 # 4
• Let A = {n | n = 5r for some integer r} and B = {m | m = 20s for some integer s}.
a. Is A B? Explain.
b. Is B A? Explain.
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6.1 Set Theory - Definitions and the Element Method of Proof
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Set Equality
• Given sets A and B, A equals B, written A = B, iff every element of A is in B and every element of B is in A. Symbolically,
A = B A B and B A
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6.1 Set Theory - Definitions and the Element Method of Proof
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Operations on Sets
• Let A and B be subsets of a universal set U.
1. The union of A and B denoted A B, is the set of all elements that are in at least one of A or B.
Symbolically: A B = {x U | x A or x B}
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6.1 Set Theory - Definitions and the Element Method of Proof
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Operations on Sets
• Let A and B be subsets of a universal set U.
2. The intersection of A and B denoted A B, is the set of all elements that are common to both A or B.
Symbolically: A B = {x U | x A and x B}
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6.1 Set Theory - Definitions and the Element Method of Proof
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Operations on Sets
• Let A and B be subsets of a universal set U.
3. The difference of B minus A (or relative complement of A in B) denoted B – A, is the set of all elements that are in B but not A.
Symbolically: B – A = {x U | x B and x A}
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6.1 Set Theory - Definitions and the Element Method of Proof
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Operations on Sets
• Let A and B be subsets of a universal set U.
4. The complement of A denoted Ac, is the set of all elements in U that are not A.
Symbolically: Ac = {x U | x A}
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6.1 Set Theory - Definitions and the Element Method of Proof
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Example – pg. 350 # 11
• Let the universal set be the set R of all real numbers and let
A = {x R | 0 < x 2}, B = {x R | 1 x < 4}, and C = {x R | 3 x < 9}. Find each of the following:
a. A B b. A B c. Ac
d. A C e. A C f. Bc
g. Ac Bc h. Ac Bc i. (A B)c
j. (A B)c
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6.1 Set Theory - Definitions and the Element Method of Proof
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Unions and Intersections of an Indexed Collection of Sets
Given sets A0, A1, A2, … that are subsets of a universal set U and given a nonnegative integer n,
0
0
0
0
| for at least one 0,1,2,...,
| for at least one nonnegative integer
| for all 0,1,2,...,
| for all nonnegative integers
n
i ii
i ii
n
i ii
i ii
A x U x A i n
A x U x A i
A x U x A i n
A x U x A i
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Definitions
• Empty Set
A set with no elements is called the empty set (or null set) and denoted by the symbol .
• Disjoint
Two sets are called disjoint iff they have no elements in common. Symbolically:
A and B are disjoint A B =
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Definitions
• Mutually Disjoint
Sets A1, A2, A3, … are mutually disjoint (or pairwise disjoint or nonoverlapping) iff no two sets Ai and Aj with distinct subscripts have any elements in common. More precisely, for all i, j = 1, 2, 3, …
Ai Aj = whenever i j.
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6.1 Set Theory - Definitions and the Element Method of Proof
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Example – pg. 305 # 23
Let for all
positive integers i.
1 1 1 1| ,iV x x
i i i i
4 4
1 1
1 2 3
1 1
1 1
a. ? b. ?
c. Are , , ,... mutually disjoint? Explain.
d. ? e. ?
f. ? g. ?
i ii i
n n
i ii i
i ii i
V V
V V V
V V
V V
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6.1 Set Theory - Definitions and the Element Method of Proof
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Definition
• Partition
A finite or infinite collection of nonempty sets {A1, A2, A3, …} is a partition of a set A iff,
1. A is the union of all the Ai
2. The sets A1, A2, A3, …are mutually disjoint.
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6.1 Set Theory - Definitions and the Element Method of Proof
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Example – pg. 351 # 27
b. Is , , , , , , , a partition of
, , , , , , , ?
c. Is 5, 4 , 7, 2 , 1,3, 4 , 6,8 a partition of
1, 2,3, 4,5,6,7,8 ?
e. Is 1,5 , 4,7 , 2,8,6,3 a partition of
1, 2,3, 4,5,6,7,8 ?
w x v u y q p z
p q u v w x y z
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6.1 Set Theory - Definitions and the Element Method of Proof
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Definition
• Power Set
Given a set A, the power set of A is denoted (A), is the set of all subsets of A.
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6.1 Set Theory - Definitions and the Element Method of Proof
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Example – pg. 351 # 31
• Suppose A = {1, 2} and B = {2, 3}. Find each of the following:
a. b.
c. d.
A B A
A B A B
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