DEFINE SETS AND KINDS OF SETS
About "Define sets and kinds of sets"
Define sets and kinds of sets :
Here we are going to see definition of sets and their types.
Definition of set :
A set is a collection of well defined objects. The objects of a set are called elements or members of the set.
The main property of a set is that it is well defined. This means that given any object, it must be clear whether that object is a member (element) of the set or not.
Generally, sets are named with the capital letters A, B, C, etc. The elements of a set are denoted by the small letters a, b, c, etc.
Define sets and kinds of sets :
The concept of set is vital to mathematical thought and is being used in almost every branch of mathematics. In mathematics, sets are convenient because all mathematical structures can be regarded as sets.
Here, we are going to see the different kind of sets.
(1) Empty set (or) null set
(2) Finite set
(3) Infinite set
(4) Singleton set
(5) Equal sets
(6) Equivalent sets
(7) Subset
(8) Proper subset
(9) Power set
(10) Super set
Empty set or Null set
A set containing no elements is called the empty set or null set or void set.
Reading notation :
So, it is denoted by { } or ∅
For example,
Consider the set A = { x : x < 1, x ∈ N }
There is no natural number which is less than 1.
Therefore, A = { }, n(A) = 0.
Note : The concept of empty set plays a key role in the study of sets just like the role of the number zero in the study of number system.
Let us look into the next type of set on "Define sets and kinds of sets".
Finite set
If the number of elements in a set is zero or finite, then the set is called a finite set.
For example,
(i) Consider the set A of natural numbers between 8 and 9.
There is no natural number between 8 and 9.
So, A = { } and n(A) = 0.
Hence, A is a finite set.
(ii) Consider the set X = { x : x is an integer and -1 ≤ x ≤ 2 }
So, X = { -1, 0, 1, 2 } and n(X) = 4
Hence, X is a finite set.
Note : The cardinal number of a finite set is finite.
Infinite set
A set is said to be an infinite set if the number of elements in the set is not finite.
For example,
Let W = The set of all whole numbers .
That is, W = { 0, 1, 2, 3, ......................}
The set of all whole numbers contain infinite number of elements.
Hence, W is an infinite set.
Note : The cardinal number of an infinite set is not a finite number.
Let us look into the next type of set on "Define sets and kinds of sets".
Singleton set
A set containing only one element is called a singleton set.
For example,
Consider the set A = { x : x is an integer and 1 < x < 3 }
So, A = { 2 }. That is, A has only one element.
Hence, A is a singleton set.
Note : { 0 } is not null set. Because it contains one element. That is "0".
Equivalent Set
Two sets A and B are said to be equivalent if they have the same number of elements.
In other words, A and B are equivalent if n(A) = n(B).
Reading notation :
"A and B are equivalent" is written as A ≈ B
For example,
Consider A = { 1, 3, 5, 7, 9 } and B = { a, e, i, o, u }
Here n(A) = n(B) = 5
Hence, A and B are equivalent sets.
Equal sets
Two sets A and B are said to be equal if they contain exactly the same elements, regardless of order.
Otherwise the sets are said to be unequal.
In other words, two sets A and B are said to be equal if
(i) every element of A is also an element of B and
(ii) every element of B is also an element of A.
Reading notation :
For example,
Consider A = { a, b, c, d } and B = { d, b, a, c }
Set A and set B contain exactly the same elements.
And also n(A) = n(B) = 4.
Hence, A and B are equal sets.
Note : If n(A) = n(B), then the two sets A and B need not be equal. Thus, equal sets are equivalent but equivalent sets need not be equal.
Subset
A set X is a subset of set Y if every element of X is also an element of Y.
In symbol we write
x ⊆ y
Reading Notation :
Read ⊆ as "X is a subset of Y" or "X is contained in Y"
Read ⊈ as "X is a not subset of Y" or "X is not contained in Y"
Proper subset
A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.
In symbol, we write X ⊂ Y
Reading notation :
Read X ⊂ Y as "X is proper subset of Y"
The figure given below illustrates this.
Power set
The set of all subsets of A is said to be the power set of the set A.
Reading notation :
The power set of A is denoted by P(A)
Super set
A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.
In symbol, we write X ⊂ Y
Here,
Y is called super set of X
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