Transcript 6.1 Set Theory: Definitions & the Element Method of Proof

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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Subsets
• The negation is existential
A
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B  x, if x  A and x  B.
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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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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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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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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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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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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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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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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
d. A C
g. Ac Bc
j. (A B)c
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b. A
e. A
B
C
h. Ac
c. Ac
f. Bc
Bc
6.1 Set Theory - Definitions and the
Element Method of Proof
i. (A
B)c
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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,
n
Ai   x  U | x  Ai for at least one i  0,1, 2, ..., n 
i0

Ai   x  U | x  Ai for at least one nonnegative integer i 
i0
n
Ai   x  U | x  Ai for all i  0,1, 2, ..., n 
i0

Ai   x  U | x  Ai for all nonnegative integers i 
i0
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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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Example – pg. 305 # 23
Let

Vi   x 

1  1 1
|   x     , 
i
i  i i
1
for all
positive integers i.
4
4
Vi  ?
a.
Vi  ?
b.
i 1
i 1
c. A re V1 , V 2 , V 3 , ... m utually disjoint? E xplain.
n
n
Vi  ?
d.
i 1
i 1


Vi  ?
f.
i 1
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Vi  ?
e.
Vi  ?
g.
i 1
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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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Example – pg. 351 # 27
b. Is
 w , x , v  ,  u , y , q  ,  p , z 
a partition of
 p, q, u , v, w, x, y, z ?
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  ?
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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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Example – pg. 351 # 31
• Suppose A = {1, 2} and B = {2, 3}. Find each
of the following:
a.   A  B 
b.   A 
c.   A  B 
d.   A  B 
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Element Method of Proof
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