CONTENT LIST
Introduction
Sets are a collection of well-defined objects or elements.
A set is represented by a capital letter symbol and the number of elements is represented in a curly bracket {…}
For example, set A is a collection of all the natural numbers, such as A = {1,2,3,4,5,6,7,8,…..∞}
Examples of standard sets
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
T : the set of all irrational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers
R+ : the set of positive real numbers
Representation of set
Sets can be represented in two forms:
- Roster Form:
Example- A= Set of even numbers less than 8
A={2,4,6} - Set Builder Form:
Example: A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4
Types of set
- Empty Set or Null set: It has no element present in it. Example: A = {} is a null set.
- Finite Set: It has a limited number of elements. Example: A = {1,2,3,4}
- Infinite Set: It has an infinite number of elements. Example: A = {x: x is the set of all whole numbers}
- Equal Set: Two sets which have the same members. Example: A = {1,2,5} and B={2,5,1}: Set A = Set B
- Equivalent Set: Two sets which have equal no. of members. Example: A = {1,2,5} and B={4,6,8}: Set A is equivalent of Set B
- Subsets: A set ‘A’ is said to be a subset of B if each element of A is also an element of B . Example: A={1,2}, B={1,2,3,4}, then A ⊆ B
- Universal Set: A set which consists of all elements of other sets present in a Venn diagram . Example: A={1,2}, B={2,3}, The universal set here will be, U = {1, 2,3}
- Singleton Set: A set which contains a single element is called a singleton set.
- Disjoint Sets: The two sets A and B are said to be disjoint if the set does not contain any common element.
Order of set
The order of a set defines the total number of elements in a set. It describes the size of a set. The order of set is also known as the cardinality.
The size of set whether it is a finite set or an infinite set, said to be set of finite order or infinite order, respectively.
Subset
Set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B. Even the null set is considered to be the subset of another set.
The set is also a subset of itself.
Number of Subsets of a given Set:
If a set contains ‘n’ elements, then the number of subsets of the set is 2^n.
Proper Subset
If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B.
Example: If A = {2,5,7} is a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7}
But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.
Number of Proper Subsets of the Set
If a set contains ‘n’ elements, then the number of proper subsets of the set is (2^n - 1) Where n= cordinality of set.
If A = {p, q} the proper subsets of A are [{ }, {p}, {q}]
Number of proper subsets of A are = 2² - 1 = 4 – 1 = 3
Super set
Set A is said to be the superset of B if all the elements of set B are the elements of set A. It is represented as A ⊃ B.
For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of B
Universal Set
A universal set (usually denoted by U) is a set which has elements of all the related sets, without any repetition of elements.
Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be:
U = {1,2,3,4,5}
Operations on Sets
- Union of sets
- Intersection of sets
- A complement of a set
- Set difference
- Cartesian product of sets
Union of sets
If set A and set B are two sets, then A union B is the set that contains all the elements of set A and set B. It is denoted as A ∪ B.
Example: Set A = {1,2,3} and B = {4,5,6}, then A union B is:
A ∪ B = {1,2,3,4,5,6}
Intersection of sets
If set A and set B are two sets, then A intersection B is the set that contains only the common elements between set A and set B. It is denoted as A ∩ B.
Example: Set A = {1,2,3} and B = {4,5,6}, then A intersection B is:
A ∩ B = { } or Ø
Since A and B do not have any elements in common, so their intersection will give null set.
A complement of a set
The complement of any set, say P, is the set of all elements in the universal set that are not in set P. It is denoted by P’.
Properties of Complement sets
P ∪ P′ = P
P ∩ P′ = Φ
Set difference
If set A and set B are two sets, then set A difference set B is a set which has elements of A but no elements of B. It is denoted as A – B.
Example: A = {1,2,3} and B = {2,3,4}
A – B = {1}
Cartesian Product of sets
If set A and set B are two sets then the Cartesian product of set A and set B is a set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. It is denoted by A × B.
We can represent it in set-builder form, such as:
A × B = {(a, b) : a ∈ A and b ∈ B}
Example: set A = {1,2,3} and set B = {Bat, Ball}, then;
A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}
Intervals of sets
An interval is a set of real numbers that contains all real numbers lying within any two specific numbers of the set R.
Intervals is the subsets of real numbers.
Let us consider an interval as 2 < x < 7, which means a set of numbers lying between 2 and 7 (excluding 2 and 7), this represent the value of x.
Types of Interval
- Open intervals
- Closed intervals
- Right half open intervals
- Left half open intervals
- Degenerate intervals
Open Interval
The set of real numbers {x : a < x < b} is called an open interval and is denoted by (a, b).
Open intervals (a, b) is having all the points between a and b, except a and b.
Closed Interval
The interval is having the endpoints is also called the closed interval and is denoted by [a, b]. It is written as [a, b] = {x : a ≤ x ≤ b}.
Right Half Open Interval
Right half open intervals mean the intervals that are closed at left hand side and open at right hand side. This can be represented as:
[a, b) = {x : a ≤ x < b} is an right open interval from a to b, including a but excluding b.
Left Half Open Interval
Left half open intervals mean the intervals that are closed at right hand side and open at left hand side. This can be represented as:
(a, b] = {x : a < x ≤ b} is an left open interval from a to b, including b but excluding a.
Degenerate Interval
A degenerate interval is the set consisting of any single real number.
Example: [a, a] is an example of degenerate interval because this interval is having only one real number.
