SETS
01 Introduction
03
A set is collection of well-defi-
ned distinguished objects. 02
By well defined we mean that Cardinal Number
there should be no ambiguity The number of elements in a finite set Let A and B
regarding the inclusion and is represented by n(A), known as element o
exclusion of the objects. cardinal number. written as A
For example a collection of Eg .: A = {a, b, c, d, e} Then, n(A) = 5 or 'B' co
scariest movies can’t be consi- Note:
• Every set is
dered as a set, because it will • If A is not a
differ from person to person • The empty
• If A is a set
A are 2n an
2n-1
04 Types of Sets 05 Operations on Sets
Eg. Let A={3,
Here, n(A) =
Empty set or Null set Difference of two sets
If A and B are two sets, then their
A set which has no element is difference A-B is defined as:
called null set. It is denoted by A-B={x : x ∈A and x∉B} Similarly,
symbol φ or {}. B-A={x:x∈B and x ∉A}
Equivalent set
Two finite sets A and B are said to Union
be equivalent, if n(A) = n(B). The union of two sets A and B, written
Clearly, equal set are equivalent as A∪B (read as A union B) is the set of
all elements which are either in A or in B
but equivalent set need not to be or in both. Thus, A∪B={x : x ∈A or x ∈B}
equal.
A B
Singleton set
A set having one element is called
singleton set. clearly, x ∈ A∪B ⇒ x ∈A or x ∈B and x ∉A
∪B ⇒ x ∉A and x ∉B
Finite and Infinite set
A set which has finite number of Symmetric Difference
elements is called a finite set. The symmetric difference of two
Otherwise, it is called an infinite sets A and B, denoted by A∆B, in
set. defined as (A∆B)=(A-B)∪(B-A)
Power set Disjoint sets
The set of all subset of a given set
A is called power set of A and U
denoted by P(A). A B (i)
Two sets A and B are said to be
Equal set disjoint, if A∩B = ϕ i. e, A and B
Two sets A and B are said to be have no common element.
equal, written as A=B, if every
element of A is in B and every
element of B is in A.
07
06
Results on Operation of Sets: Ca
A×B=
1. A ⊆ A ∪ B, B ⊆ A ∪ B, A ∩ B ⊆ A, A ∩ B ⊆ B.
01 Introduction
03
A set is collection of well-defi-
ned distinguished objects. 02
By well defined we mean that Cardinal Number
there should be no ambiguity The number of elements in a finite set Let A and B
regarding the inclusion and is represented by n(A), known as element o
exclusion of the objects. cardinal number. written as A
For example a collection of Eg .: A = {a, b, c, d, e} Then, n(A) = 5 or 'B' co
scariest movies can’t be consi- Note:
• Every set is
dered as a set, because it will • If A is not a
differ from person to person • The empty
• If A is a set
A are 2n an
2n-1
04 Types of Sets 05 Operations on Sets
Eg. Let A={3,
Here, n(A) =
Empty set or Null set Difference of two sets
If A and B are two sets, then their
A set which has no element is difference A-B is defined as:
called null set. It is denoted by A-B={x : x ∈A and x∉B} Similarly,
symbol φ or {}. B-A={x:x∈B and x ∉A}
Equivalent set
Two finite sets A and B are said to Union
be equivalent, if n(A) = n(B). The union of two sets A and B, written
Clearly, equal set are equivalent as A∪B (read as A union B) is the set of
all elements which are either in A or in B
but equivalent set need not to be or in both. Thus, A∪B={x : x ∈A or x ∈B}
equal.
A B
Singleton set
A set having one element is called
singleton set. clearly, x ∈ A∪B ⇒ x ∈A or x ∈B and x ∉A
∪B ⇒ x ∉A and x ∉B
Finite and Infinite set
A set which has finite number of Symmetric Difference
elements is called a finite set. The symmetric difference of two
Otherwise, it is called an infinite sets A and B, denoted by A∆B, in
set. defined as (A∆B)=(A-B)∪(B-A)
Power set Disjoint sets
The set of all subset of a given set
A is called power set of A and U
denoted by P(A). A B (i)
Two sets A and B are said to be
Equal set disjoint, if A∩B = ϕ i. e, A and B
Two sets A and B are said to be have no common element.
equal, written as A=B, if every
element of A is in B and every
element of B is in A.
07
06
Results on Operation of Sets: Ca
A×B=
1. A ⊆ A ∪ B, B ⊆ A ∪ B, A ∩ B ⊆ A, A ∩ B ⊆ B.
