Post on 22-Jan-2017
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August 2006 1
August 2006 2
Definitions
• A set is a collection of objects.
• Objects in the collection are called elements of the set.
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Examples - set The collection of persons living in INDIA is
a set.– Each person living in INDIA is an element of
the set.
The collection of all countries in the state of Texas is a set.– Each county in Texas is an element of the set.
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Examples - set The collection of counting numbers is a
set.– Each counting number is an element of the
set.
The collection of pencils in your briefcase is a set.– Each pencil in your briefcase is an element
of the set.
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Notation • Sets are usually designated with
capital letters.
• Elements of a set are usually designated with lower case letters.
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Notation• The roster method of specifying a
set consists of surrounding the collection of elements with braces.
For example the set of counting numbers from 1 to 5 would be written as
{1, 2, 3, 4, 5}.
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Notation• Set builder notation has the general
form
{variable | descriptive statement }. The vertical bar (in set builder notation) is always
read as “such that”.
Set builder notation is frequently used when the roster method is either inappropriate or inadequate.
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Example – set builder notation
{x | x < 6 and x is a counting number} is the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}.
{x | x is a fraction whose numerator is 1 and whose denominator is a counting number }.
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Definition• The set with no elements is called
the empty set or the null set and is designated with the symbol .
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Examples – empty set– The set of all pencils in your briefcase might indeed be the empty set.
– The set of even prime numbers greater than 2 is the empty set.
– The set {x | x < 3 and x > 5} is the empty set.
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Definition - subset• The set A is a subset of the set B
if every element of A is an element of B.
• If A is a subset of B and B contains elements which are not in A, then A is a proper subset of B.
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Notation - subsetIf A is a subset of B we write A B to designate that relationship.
If A is a proper subset of B we write A B to designate that relationship.
If A is not a subset of B we write A B to designate that relationship.
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Example - subset The set A = {1, 2, 3} is a subset of the set B
={1, 2, 3, 4, 5, 6} because each element of A is an element of B.
We write A B to designate this relationship between A and B.
We could also write {1, 2, 3} {1, 2, 3, 4, 5, 6}
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Example - subset The set A = {3, 5, 7} is not a subset
of the set B = {1, 4, 5, 7, 9} because 3 is an element of A but is not an element of B.
The empty set is a subset of every set, because every element of the empty set is an element of every other set.
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PROPER SUBSET:
• If A⊆B and A≠B then A is called a proper subset of B, written as A⊂B.
• For e.g.A={1,3,5},B={1,2,3,4,5,6}
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UNIVERSAL SET:• A set U is called a universal set. If all the sets
under consideration are sub-sets of the set U.• For e.g. If A {1,2,3,4},B {2,3,5,7} and
C={2,4,6,8}then the universal set U={1,2,3,4,5,6,7,8,9}.
POWER SET:• The collection of all possible subsets of a
given set A is the power set of A. It is denoted by P(A).
• For e.g. If A={1,2,3} then• P(A)={Ø,{1},{2},{3},{1,2},{2,3},
{1,3},{1,2,3}}.
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EQUAL SETS• Two sets A and B are equal if A
B and B A. If two sets A and B are equal we write A = B to designate that relationship.
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Example
The sets A = {3, 4, 6} and B = {6, 3, 4} are equal because A B and B A.The definition of equality of sets shows that the order in which elements are written does not affect the set.
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ExampleIf A = {1, 2, 3, 4, 5} and B = {x | x < 6 and x is a counting number} then A is a subset of B because every element of A is an element of B and B is a subset of A because every element of B is an element of A.
Therefore the two sets are equal and we write A = B.
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Definition - union• The union of two sets A and B is the set
containing those elements which are elements of A or elements of B. We write A BIf A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} then A B = {1, 2, 3, 4, 5, 6}.
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Example - UnionIf A is the set of prime numbers and B is the set of even numbers then A B = {x | x is even or x is prime }.
If A = {x | x > 5 } and B = {x | x < 3 } then A B = {x | x < 3 or x > 5 }.
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Venn Diagram - unionA is represented by the red circle and B isrepresented by the blue circle.The purple colored regionillustrates the intersection.The union consists of allpoints which are colored red or blue or purple.
A B
A∩B
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Definition - intersection• The intersection of two sets A
and B is the set containing those elements which are
elements of A and elements of B.
We write A B
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ExampleIf A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then A B = {3, 6}If A is the set of prime numbers and B is the set of even numbers then A ∩ B = { 2 }
If A = {x | x > 5 } and B = {x | x < 3 } then A ∩ B =
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ExampleIf A = {x | x < 4 } and B = {x | x >1 } thenA ∩ B = {x | 1 < x < 4 }
If A = {x | x > 4 } and B = {x | x >7 } thenA ∩ B = {x | x < 7 }
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These sets can be visualized with circles in what is called a Venn Diagram.
A B
A BEverything that is in
A AND B.
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• Disjoint sets:Two sets A and B are disjoint sets, if A∩B=Ø.For e.g.
If A={2,4,6,8} and B={ 1,3,5,7} then A and B are disjoint sets.
Note: If A∩B ≠Ø, then A and B are said to be intersecting sets or overlapping sets.
• Difference of sets: If A and B are two sets, then their difference A-B is the set containing exactly those element in A that are not in B.
Eg:- A={a,b,c} B={a,d} A-B={b,c}
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• Symmetric difference:-Symmetric difference of two sets P&Q is the set containing exactly all the elements that are not in P or in Q but not in both.
P(+)Q=(PUQ)-(P ∩Q) Eg:-P={a,b} Q={a,c} Then symmetric difference is
{b,c}
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• Union and intersection are commutative operations.
A B = B A A ∩ B = B ∩ A• Union and intersection are associative
operations. (A B) C = A (B C) (A ∩ B) ∩ C = B ∩ (A ∩ C)
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• Idempotent law:- AUA=A A∩A=A• Identity law:- AU=A AUU=U A∩U=A• Complement law:- AUA’=U U’=
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• Two distributive laws are true.
A ∩ ( B C )= (A ∩ B) (A ∩ C) A ( B ∩ C )= (A B) ∩ (A C)