**Finite and infinite sets :**** **

Here, we are going to see the important basic stuff finite and infinite sets on set theory.

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.

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.

Apart from the stuff "Finite and infinite sets", let us look at the other types of sets in set theory.

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.

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".

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.

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.

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"

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.

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)

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 **

After having gone through the stuff given above, we hope that the students would have understood "Finite and infinite sets".

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