1. Denoting the beginning
SET THEORY
What is a set?
A set is a collection of distinct objects. The objects in a set are called the elements or the members of the set. The name of the set is written in upper case and the elements of the set are written in lower case. If x is an element of a set A, we say that x belongs to or is a member of A, and is expressed symbolically as
x A.
If y is not a member of A, then this is symbolically denoted as
y A
Let V be the set of all vowels. Then V is written as
V= {a, e, i, o, u}
Name of the
set
The curly brackets
Denoting the beginning
And end
The definition of a set should be such that given an element, you can unambiguously judge whether that
Kavita Hatwal Fall 2002

2. The order in which elements appear do not matter, so
{a,o,i,e, u}, {u, e, o, a, i}, {o, e, a, i,u} are all V
a {a}
a is an element whereas {a} is a set whose element is a. {} makes all the difference in set notation.
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Sets given by defining property.
{x R | -2 < x < 5 }
Is read as (from left to right) the set of all x such that x is a real number and also x is greater than –2 and less than 5. Can you list some of the members of this set?
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Finite and infinite sets.
Sets whose elements can be listed are called finite sets, like
H={seasons in a year}
Which of the above sets are finite and which are infinite?
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3. Can you find any x which satisfies the above condition?
SUBSETS
If A and B are sets , A is called the subset of B, written as A B, if and only if, every element of A is also an element of B.
Symbolically,
A B , x, if x A then x B.
Also A is contained in B or B contains A are ways of saying that A is a subset of B.
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EMPTY SET
Consider the set X=
Can you find any x which satisfies the above condition?
A = {set of all animals}
P={x A| x is a pink elephant}
What could possibly be the elements of P?
A set with no elements is called an empty set denoted as An empty
Set is a subset of every set A, where A is any set.
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4. Is in B but there is at least one element of B that is not in A
because is the set with no elements {}, whereas { } is the set with one element, the empty set.
Subsets revisited
Let A and B be sets. A is a proper subset of B if, and only if, every element of A
Is in B but there is at least one element of B that is not in A
For every set A
A A, i.e. each set is its own subset
And
A, i.e. the empty set is subset of every set
U is the universal set or universe of discourse. It is considered the all
encompassing set. Every set is a subset of U.
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5. For example, if A and B are sets and
Venn Diagrams
What is a Venn Diagram?
A Venn Diagram is a pictorial representation of sets. U is the universal set which is represented as a rectangle. Other sets are represented as circles.
For example, if A and B are sets and
A B, that is B contains A, then this situation is represented as
More on Venn Diagrams later
Z = set of integers
Q = set of rational numbers.
R= set of real numbers
Can you show the subset relation between Z, Q and R using notation and Venn Diagram notation ? A B
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6. element of A is in B and every element of B is in A. Symbolically,
Set Equality
Venn Diagram
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Given two sets A and B, A equals B, written A = B, if, and only if, every
element of A is in B and every element of B is in A.
Symbolically,
A=B
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7. Let A and B be subsets of the universal set U.
Set Operations.
Let A and B be subsets of the universal set U.
The Union of A and B , denoted , is the set of all elements x in U such
that x is either in A or in B.
2. The Intersection of A and B , denoted , is the set of all elements x in U
such that x is in both A and B.
3. The Difference of B minus A , denoted B-A, is the set of all elements x in U
such that x is in B, but not in A.
The complement of a set A, denoted as is the set of all elements x in U such
that x is not in A.
Symbolically
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8. Show the facts of previous slides using Venn Diagrams
Class Activity Draw the rest on your own
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9. A  B = B  A. A  B = B  A. c = U. Uc = . Complement A  Ac = .
A  Ac = U.
Distributivity
A  (B  C) = (A  B)  (A  C).
A  (B  C) = (A  B)  (A  C).
Identity
A  U = A.
A   = A.
Commutativity
A  B = B  A.
A  B = B  A.
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10. Associativity (A  B)  C = A  (B  C). (A  B)  C = A  (B  C).
Idempotent
A  A = A.
A  A = A.
Universal Bounds
A   = .
A  U = U.
Two sets A and B are called disjoint if they have no elements in common, i.e.
A  B = 
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Kavita Hatwal Fall 2002

11. Given 2 sets A and B, the Cartesian Product of A and B , denoted
, is the set of all ordered
pairs(a,b), where a is in A and b is in B.
Given sets the Cartesian Product of denoted by
is the set of all ordered n-tuples
where
Symbolically,
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Given a set A, the power set of A, denoted P (A), is the set of all subsets of A.
For all sets A and B, then 1. 2.
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Kavita Hatwal Fall 2002

12. Two sets A and B are disjoint if A  B = .
Sets A1, …, An are pairwise disjoint if Ai  Aj =  for all i, j  {1, …, n}, with i  j.
A collection of sets {A1, …, An} is a partition of a set A if
A1  …  An = A, and
A1, …, An are pairwise disjoint.
Kavita Hatwal Fall 2002

13. Then {A0, A1, A2} is a partition of Z.
Let
A0 = {n  Z | n = 3k for some k  Z}.
A1 = {n  Z | n = 3k + 1 for some k  Z}.
A2 = {n  Z | n = 3k + 2 for some k  Z}.
Then {A0, A1, A2} is a partition of Z.
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14. Set Identities
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15. A Boolean algebra is a set S that includes two elements 0 and 1,
Boolean Algebras
A Boolean algebra is a set S that
includes two elements 0 and 1,
has two binary operations + and ,
has one unary operation ,
which satisfy the following properties for all a, b, c in S:
Kavita Hatwal Fall 2002

16. Boolean Algebras Commutativity a + b = b + a. a  b = b  a.
Associativity
(a + b) + c = a + (b + c).
(a  b)  c = a  (b  c).
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17. Boolean Algebras Distributivity a  (b + c) = (a  b) + (a  c)
Identity
a + 0 = a.
a  1 = a.
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18. Boolean Algebras Complementation a + a = 1. a  a = 0.
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19. Examples: Boolean Algebras
Let U be a nonempty universal set. Let 0 be  and 1 be U. Let + be  and  be . Let  be complementation. Then U is a Boolean algebra.
Let U be a nonempty universal set. Let 0 be U and 1 be . Let + be  and  be . Let  be complementation. Then U is a Boolean algebra.
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20. Examples: Boolean Algebras
Let S be the set of all statements. Let 0 be F and 1 be T. Let + be  and  be . Let  be negation. Let = be . Then S is a Boolean algebra.
Let S be the set of all statements. Let 0 be T and 1 be F. Let + be  and  be . Let  be negation. Let = be . Then S is a Boolean algebra.
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21. Theorem: Let S be a Boolean algebra and let a, b in S. Then a  a = a.
Derived Properties
Theorem: Let S be a Boolean algebra and let a, b in S. Then
a  a = a.
a + a = a.
a  0 = 0.
a + 1 = 1.
a  b = a if and only if a + b = b.
a  b = a + b if and only if a = b.
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22. Derived Properties, continued
a  b = 0 and a + b = 1 if and only if a = b.
0 = 1.
1 = 0.
(a) = a.
(a  b) = a + b.
(a + b) = a  b.
Kavita Hatwal Fall 2002

23. Example: Boolean Algebra
Let S = {1, 2, 3, 5, 6, 10, 15, 30}.
Define
a + b = gcd(a, b).
a  b = lcm(a, b).
a = 30/a.
1 is U, 30 is 0
Verify the 10 basic properties.
Kavita Hatwal Fall 2002