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Sets
Section 6.1 Basic Definitions of Sets Theory
Section 6.2 Properties of sets
Section 6.3 Proofs and Boolean Algebras
The most fundamental notion in all of mathematics is that of a set.
We say that a set is a specified collection of objects, called
elements (or members) of the set.
We denote sets by capital letters A, B, … and elements by lower
case letters, like x, y , … and so on. If an element x belongs to a set
A, we denote this by x  A, if not we write x A.
Instructor: Hayk Melikya
Introduction to Abstract Mathematics
[email protected]
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Specifying set
There are various ways to specify a set. For the set of natural
numbers less than or equal to 5, you could write {1, 2, 3, 4, 5} .
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For sets that cannot be specified by a list, we describe the
elements by some property common to the elements in the
set but no others, such as in the description
A = { x | P (x)}
which reads “the set of all x such that the condition P(x) is
true.”
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Common Sets:
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Subsets
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We say that a set A is a subset of a set B if every element of A is also an
element of B.
Symbolically, we write this as A  B and is read “A is contained in B.”
Finally, the notation A  B means that A is not a subset of B.
Sets are often illustrated by Venn diagrams, where sets are represented
as circles and elements of the set are points inside the circle.
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Equality of Sets:
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Two sets are equal (A = B) if they consist of exactly the same elements.
In other words, they are equal if
(A = B) if and only if (x) (( xA  x B)
or
(A = B) iff (x) (( xA  x B)  (xB  x A ))
another way:
(A = B) if and only if (A  B  B  A).
Empty Set: The set with no elements is called the empty set (or null set) and
denoted by the Greek letter  (or sometimes the empty bracket { } )
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Theorem 1 (Guaranteed Subset)
For any set A, we have   A.
Proof :
Since the goal is to show
xxA
our job is done before we begin.
The reason being that the hypothesis x   of the implication is false, being
that  contains no elements, hence the proposition is true regardless of the
set A. In other words  is a subset of any set.
END
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Theorem 2 (Transitive Subsets) Let A, B and C be sets.
If A  B and B  C then A  C .
Proof:
We will prove the conclusion A  C and use the hypothesis as needed.
Letting x  A the goal is to show x  C . Since x  A and using the
assumption A  B , we know x  B . But the second hypothesis says
B  C , and so we know x  C . Hence, we have proved A  C , which
proves the theorem.
END
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Subset and Membership:
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Power Set P(A)
An important set in mathematics is the power set.
For every set A, we denote by P(A) the set of all subsets of A.
Theorem 3 (Power Set) Let A and B be sets.
Then A  B if and only if P (A)  P(B).
Proof:
(A  B)  (P(A)  P(B)):
We start by letting X  P(A) and show X  P(B) (and use A  B as
our “helper”). Letting X  P(A) we have X  A and hence X  B. But
this means X  P(B) and so we have shown P(A)  P(B) .
(P(A)  P(B)) (A  B) :
We let x  A and show x  B . If x  A, then {x }  P(A) , and since
P(A)  P(B) we know {x }  P(B) . But this means x  B and so A  B .
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Sec 6.2 Operations on Sets Union, Intersection and
Complement
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In traditional arithmetic and algebra, we carry out the binary operations of
+ and × on numbers. In logic, we have the analogous binary operations
of  and  on sentences. In set theory we have the binary operations of
union  and intersection  of sets, which in a sense are analogous to the
ones in arithmetic and sentential logic.
Definition ( Union): The union of two sets A and B, denoted
A  B , is the set of elements that belong to A or B or both.
Symbolically
A  B = {x | x  A  x  B }
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Definition ( Intersection):
The intersection of two sets A and B, denoted A  B , is the set
of elements that belong to A and B.
Symbolically
A  B = {x | x  A  x  B }
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Definition( Complement):
The compliment of A, denoted Ac is the set of elements
belonging to the universal set U but not A.
Symbolically
Ac = {x | x U  x  A } .
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Definition (Relative Complement or Difference):
The relative complement of A in B, denoted, B \ A, is the set of
elements in B but not in A.
Symbolically
B \ A = {x | x  B  x  A }
The concepts of union, intersection and relative complement of sets
can be illustrated graphically by use of Venn diagrams.
Each Venn diagram begins with an oval representing the universal
set, a set that contains all elements of in discussion.
Then, each set in the discussion is represented by a circle, where
elements belonging to more than one set are placed in sections where
circles overlap.
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Venn diagrams for two overlapping sets.
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Set Identities
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Commutative Laws: A  B = A  B and A  B = B  A
Associative Laws: (A  B)  C = A  (B  C) and (A  B)  C = A  (B  C)
Distributive Laws:
A  (B  C) = (A  B)  (A  C) and A  (B  C) = (A  B)  (A  C)
Intersection and Union with universal set: A  U = A and A  U = U
Double Complement Law: (Ac)c = A
Idempotent Laws: A  A = A and A  A = A
De Morgan’s Laws: (A  B)c = Ac  Bc and (A  B)c = Ac  Bc
Absorption Laws: A  (A  B) = A and A  (A  B) = A
Alternate Representation for Difference: A – B = A  Bc
Intersection and Union with a subset: if A  B, then A  B = A and A  B = B
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Exercises
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Is is true that (A – B)  (B – C) = A – C?
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Show that (A  B) – C = (A – C)  (B – C)
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Is it true that A – (B – C) = (A – B) – C?
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Is it true that (A – B)  (A  B) = A?
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Set Partitioning
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Two sets are called disjoint if they have no elements in common
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Theorem: A – B and B are disjoint
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A collection of sets A1, A2, …, An is called mutually disjoint when any pair of sets
from this collection is disjoint
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A collection of non-empty sets {A1, A2, …, An} is called a partition of a set A when
the union of these sets is A and this collection consists of mutually disjoint sets
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Power Set
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Power set of A is the set of all subsets of A
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Theorem: if A  B, then P(A)  P(B)
Theorem:
If set X has n elements, then P(X) has 2n elements.
Proof. (By Induction)
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Boolean Algebra
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Boolean Algebra is a set of elements together with two
operations denoted as + and * and satisfying the following
properties:
a + b = b + a, a * b = b * a
(a + b) + c = a + (b + c), (a * b) *c = a * (b * c)
a + (b * c) = (a + b) * (a + c), a * (b + c) = (a * b) + (a * c)
a + 0 = a, a * 1 = a for some distinct unique 0 and 1
a + ã = 1, a * ã = 0
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Exercises
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Simplify: A  ((B  Ac)  Bc)
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Symmetric Difference: A  B = (A – B)  (B – A)
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Show that symmetric difference is associative
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Are A – B and B – C necessarily disjoint?
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Are A – B and C – B necessarily disjoint?
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Russell’s Paradox
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Set of all integers, set of all abstract ideas
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Consider S = {A, A is a set and A  A}
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Is S an element of S?
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Barber puzzle: a male barber shaves all those men who do
not shave themselves. Does the barber shave himself?
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Consider S = {A  U, A  A}. Is S  S?
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