What is a Set?
elements present involving all the elements
->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.
Representation of a Set
A set can be represented by various methods. Following are some common methods:
1. Statement form
2. Roaster form or tabular form method
3. Set Builder method
4. Venn Diagram
1)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.
2)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 }.
2. The set of all vowels in the English alphabet.
V = { a , e , i , o , u }.
3)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}.
4)Venn Diagrams :are used to represent the groups of data into circles.John Venn, a famous logician gave the concept of diagrams in 1918 If the circles are overlapping, some elements in the groups are common.
If they are not overlapping, there is nothing common between the groups or sets of the data.
The sets are represented as circles and the circles are shown inside the rectangle which is
representing the Universal set. Universal set contains all the circles since it has all the
Types of Sets
3)Singleton Set –
A set consisting of only one element is said to be Singleton set.
For example :
Set S = {5} , M = {a} are said to be singleton since they are consists of only one
element 5 and ‘a’ respectively.
4)Finite Set –
A set whose number of elements are countable i.e. finite or a set whose cardinality is a
natural number (∈ N) is said to be Finite set.
For example : Sets
A = {a, b, c, d}, B = {5,7,9,15,78} and C = { x : x is a multiple of 3, where 0<x<100)
Here A, B and C all three contain a finite number of elements i.e. 4 in A, 5 in B and 33 in C
and therefore will be called finite sets.
5)Infinite Set –
A set containing infinite number of elements i.e. whose cardinality can not found is said to
be an Infinite set.
Thus, the set of all natural numbers.
N = {1, 2, 3, 4 . . . .} is an infinite set.
Similarly, the set of all rational numbers between any two numbers will be infinite. For
example,
A = {x : x ∈ Q, 2 < x < 5} is an infinite set.
6)Equal Sets –
When two sets consists of same elements, whether in the same order, they are said to be
equal.
In other words, if each element of the set A is an element of the set B and each element of B
is an element of A, the sets A and B are called equal, i.e., A = B.
For example, A = {1,2,3,4,5} and B = {1,5,2,4,3} , then A = B.
7)Power Set –
The set of all subsets of a given set A, is said to be the power set of A.
The power set of A is denoted by P(A).
If the set A= {a, b, c}
then its subsets are ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c}.
Therefore, P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c} }.
|P(A)| = 2^n
EXAMPLES:
Q)In a school, every student plays either football or soccer. It was found that 200
students played football, 150 students played soccer and 100 students played
both. Find how many students were there in the school?
->Solution: Let us represent the number of students who played football as n(F) and the
number of students who played soccer as n(S).
n(F) = 200, n(S) = 150 and n(F ∩ S) = 100. We know that,
n(F∪S) = n(F) + n(S) − n(F∩S)
Therefore, n(F∪S)=(200+150)−100
n(F∪S) = 350 − 100 = 250
Q)A large software development company employs 100 computer programmers. Of
them, 45 are proficient in Java, 30 in C#, 20 in Python, 6 in C# and Java, 1 in Java
and Python, 5 in C# and Python, and just 1 programmer is proficient in all three
languages above.
Determine the number of computer programmers that are not proficient in any of
these three languages.
-->|U| = 100 , |J| = 45,|C| = 30,|P| = 20
|J ∩ C| = 6, |J ∩ P| = 1, |C ∩ P| = 5, |J ∩ C ∩ P| = 1
|U| = 100 , |J| = 45,|C| = 30,|P| = 20
|J ∩ C| = 6,
|J ∩ P| = 1,
|C ∩ P| = 5,
|J ∩ C ∩ P| = 1