set theory-1
A Set is an
unordered collection of objects, known as elements or members of the set.
An element ‘a’ belong to a set A can be written as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A.
An element ‘a’ belong to a set A can be written as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A.
Representation
of a Set
A set can be represented by various methods. 3 common methods used for representing set:
1. Statement form.
2. Roaster form or tabular form method.
3. Set Builder method.
A set can be represented by various methods. 3 common methods used for representing set:
1. Statement form.
2. Roaster form or tabular form method.
3. Set Builder method.
Statement form
In this representation, the well-defined description of the elements of the set is given. Below are some examples of the same.
1. The set of all even number less than 10.
2. The set of the number less than 10 and more than 1.
In this representation, the well-defined description of the elements of the set is given. Below are some examples of the same.
1. The set of all even number less than 10.
2. The set of the number less than 10 and more than 1.
Roster form
In this representation, elements are listed within the pair of brackets {} and are separated by commas. Below are two examples.
1. Let N is the set of natural numbers less than 5.
N = { 1 , 2 , 3, 4 }.
In this representation, elements are listed within the pair of brackets {} and are separated by commas. Below are two examples.
1. Let N is the set of natural numbers less than 5.
N = { 1 , 2 , 3, 4 }.
2. The set of all vowels in the
English alphabet.
V = { a , e , i , o , u }.
V = { a , e , i , o , u }.
Set builder
form
In Set-builder set is described by a property that its member must satisfy.
1. {x : x is even number divisible by 6 and less than 100}.
2. {x : x is natural number less than 10}.
In Set-builder set is described by a property that its member must satisfy.
1. {x : x is even number divisible by 6 and less than 100}.
2. {x : x is natural number less than 10}.
Equal sets
Two sets are said to be equal if both have same elements. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets.
Two sets are said to be equal if both have same elements. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets.
NOTE: Order of
elements of a set doesn’t matter.
Subset
A set A is said to be subset of
another set B if and only if every element of set A is also a part of other set
B.
Denoted by ‘⊆‘.
‘A ⊆ B ‘ denotes A is a subset of B.
Denoted by ‘⊆‘.
‘A ⊆ B ‘ denotes A is a subset of B.
To prove A is the subset of B, we
need to simply show that if x belongs to A then x also belongs to B.
To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B.
To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B.
‘U’ denotes the universal set.
Above Venn Diagram shows that A is a subset of B.
Above Venn Diagram shows that A is a subset of B.
Size of a Set
Size of a set can be finite or infinite.
Size of a set can be finite or infinite.
For example
Finite set: Set
of natural numbers less than 100.
Infinite set:
Set of real numbers.
Size of the set S is known as Cardinality
number, denoted as |S|.
Example: Let A be a set of odd
positive integers less than 10.
Solution : A = {1,3,5,7,9}, Cardinality of the set is 5, i.e.,|A| = 5.
Solution : A = {1,3,5,7,9}, Cardinality of the set is 5, i.e.,|A| = 5.
Note: Cardinality of a null set is
0.
Power Sets
The power set is the set all possible subset of the set S. Denoted by P(S).
Example: What is the power set of {0,1,2}?
Solution: All possible subsets
{∅}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}.
Note: Empty set and set itself is also the member of this set of subsets.
The power set is the set all possible subset of the set S. Denoted by P(S).
Example: What is the power set of {0,1,2}?
Solution: All possible subsets
{∅}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}.
Note: Empty set and set itself is also the member of this set of subsets.
Cardinality of
power set is
, where n is the number of elements
in a set.
Cartesian
Products
Let A and B be two sets. Cartesian product of A and B is denoted by A × B, is the set of all ordered pairs (a,b), where a belong to A and b belong to B.
Let A and B be two sets. Cartesian product of A and B is denoted by A × B, is the set of all ordered pairs (a,b), where a belong to A and b belong to B.
A × B = {(a, b)
| a ∈
A ∧
b ∈
B}.
Example 1. What is Cartesian product
of A = {1,2} and B = {p, q, r}.
Solution : A × B ={(1, p), (1, q), (1, r), (2, p), (2, q), (2, r) };
Solution : A × B ={(1, p), (1, q), (1, r), (2, p), (2, q), (2, r) };
The cardinality of A × B is N*M, where N is the Cardinality of A and M is the cardinality of B.
Note: A × B is not the same as B ×
A.
Ugc net-2
Union
Union of the sets A and B, denoted by A ∪ B, is the set of distinct element belongs to set A or set B, or both.
Union of the sets A and B, denoted by A ∪ B, is the set of distinct element belongs to set A or set B, or both.
Above is the Venn Diagram of A U B.
Example : Find the union of A = {2,
3, 4} and B = {3, 4, 5};
Solution : A ∪ B = {2, 3, 4, 5}.
Solution : A ∪ B = {2, 3, 4, 5}.
Intersection
The intersection of the sets A and B, denoted by A ∩ B, is the set of elements belongs to both A and B i.e. set of the common element in A and B.
The intersection of the sets A and B, denoted by A ∩ B, is the set of elements belongs to both A and B i.e. set of the common element in A and B.
Above is the Venn Diagram of A ∩ B.
Example: Consider the previous sets
A and B. Find out A ∩ B.
Solution : A ∩ B = {3, 4}.
Solution : A ∩ B = {3, 4}.
Disjoint
Two sets are said to be disjoint if their intersection is the empty set .i.e sets have no common elements.
Two sets are said to be disjoint if their intersection is the empty set .i.e sets have no common elements.
Above is the Venn Diagram of A
disjoint B.
For Example
Let A = {1, 3, 5, 7, 9} and B = { 2, 4 ,6 , 8} .
A and B are disjoint set both of them have no common elements.
Let A = {1, 3, 5, 7, 9} and B = { 2, 4 ,6 , 8} .
A and B are disjoint set both of them have no common elements.
Set
Difference
Difference between sets is denoted by ‘A – B’, is the set containing elements of set A but not in B. i.e all elements of A except the element of B.
Difference between sets is denoted by ‘A – B’, is the set containing elements of set A but not in B. i.e all elements of A except the element of B.
Above is the Venn Diagram of A-B.
Complement
Complement of a set A, denoted by
Complement of a set A, denoted by
, is the set of all the elements
except A. Complement of the set A is U – A.
Above is the Venn Diagram of Ac
Formula:
1.
2.
Properties of
Union and Intersection of sets:
1.
Associative
Properties: A ∪ (B ∪ C) = (A ∪ B) ∪ C
and A ∩ (B ∩ C) = (A ∩ B) ∩ C
2.
Commutative
Properties: A ∪ B
= B ∪ A and A ∩ B = B ∩ A
3.
Identity
Property for Union: A ∪ φ = A
4.
Intersection
Property of the Empty Set: A ∩ φ = φ
5.
Distributive
Properties: A ∪(B ∩ C) = (A ∪ B)
∩ (A ∪ C) similarly for intersection.
Example : Let A = {0, 2, 4, 6, 8} ,
B = {0, 1, 2, 3, 4} and C = {0, 3, 6, 9}. What are A ∪ B ∪ C and A ∩ B ∩ C ?
Solution: Set A ∪ B &cup C contains elements which are present in at
least one of A, B, and C.
A ∪ B ∪ C = {0, 1, 2, 3, 4, 6, 8, 9}.
Set A ∩ B ∩ C contains an element
which is present in all the sets A, B and C .i.e { 0 }.
Please write comments if you find
anything incorrect, or you want to share more information about the topic
discussed above
The
cardinality of the power set of {0, 1, 2 . . ., 10} is _________.
(A) 1024
(B) 1023
(C) 2048
(D) 2043
Answer: (C)
Explanation: The power set has 2n elements. For n = 11, size of power set is 2048.
(B) 1023
(C) 2048
(D) 2043
Answer: (C)
Explanation: The power set has 2n elements. For n = 11, size of power set is 2048.
The
cardinality of the power set of {0, 1, 2 . . ., 10} is _________.
(A) 1024
(B) 1023
(C) 2048
(D) 2043
Answer: (C)
Explanation: The power set has 2n elements. For n = 11, size of power set is 2048.
Q. A survey has been taken on methods of commuter travel. Each respondent was asked to check BUS, TRAIN. or AUTOMOBILE as a major method of traveling to work. More than one answer was permitted. The results reported were as follow: BUS, 30 people; TRAIN, 35 people; AUTOMOBILE, 100 people; BUS and TRAIN, 15 people; BUS and AUTOMOBILE, 15 people; TRAIN and AUTOMOBILE, 20 people; and all three methods, 5 people. How many people completed a survey form?
Solution:
The answer is given by:
30 + 35 + 100 - 15 - 15 - 20 + 5 = 120
(B) 1023
(C) 2048
(D) 2043
Answer: (C)
Explanation: The power set has 2n elements. For n = 11, size of power set is 2048.
Q. A survey has been taken on methods of commuter travel. Each respondent was asked to check BUS, TRAIN. or AUTOMOBILE as a major method of traveling to work. More than one answer was permitted. The results reported were as follow: BUS, 30 people; TRAIN, 35 people; AUTOMOBILE, 100 people; BUS and TRAIN, 15 people; BUS and AUTOMOBILE, 15 people; TRAIN and AUTOMOBILE, 20 people; and all three methods, 5 people. How many people completed a survey form?
Solution:
The answer is given by:
30 + 35 + 100 - 15 - 15 - 20 + 5 = 120
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