Types Of Sets

1) Finite & Infinite Set -

Intuitively, a set is finite if it consists of a definite number of different elements, i.e., if in counting the different numbers of the set, the counting can come to an end. A set is infinite if the process of counting its elements never comes to an end.

For example, the following are finite sets -

a) The set of students in your school.

b) The set of continents

c) The set of prime numbers between 1 and 50.

d) {x : x = 5n + 3, n ∈ Z, -2 ≤ n ≤ 2}

The following are infinite sets.

a) The set of square numbers.

b) The set of all points on a line between two distinct points A & B on it.

c)  {x : x ∈ Z, x < 0}

2) The empty set or the null set -

A set that contains no members is called the empty set or the null set.

For example, each one of the following is the empty set.

1) The set of the months of a year that have fewer than 15 days.

2) The set of students in your class who are more than 3 meters tall

3)  {x : x ∈ R, x ≠ x}

4)  {x ǀ x ∈ R, x² = - 1}

Because a real number is always equal to itself and because the square of a real number, cannot be negative, so no such real number exists.

3) The empty set is written as { } or ϕ -

There is only one empty set and it is called “the empty set” there is nothing like “an empty set” or “empty sets”

{ 0 } and { ϕ } are not empty sets because each of these sets contains one element.

4) Singleton Set

A set containing only one element is called a singleton set , e.g., A = {2} is a singleton set

5) Equal Sets

Two sets A & B are called identical or equal sets, written as A = B, if they have exactly the same elements.

For Example –

(1) if A = {p, q, r}, and B = {r, p, q}, then A = B, if they have exactly the same elements.

(2) If P = {x ǀ x² - 5x + 6 = 0}, Q = {2, 3}, R = {3, 2}, then P = Q = R

6) Cardinal Number

The number of elements in a finite set is called the cardinal number of the set. The cardinal number of a set A is abbreviated as n(A). n(A) is a symbol for cardinal number of set A.

For Example –

(1) If A = {a}, B = {a, b}, C = {p, q, r}, then n(A) = 1, n(B) = 2, n(C) = 3

(2) Let, P be the set of letters in the word “MATHEMATICS”, then A = [M, A, T, H, E, I, C, S], and n(P) = 8

a) Cardinal number of an infinite set is not defined.

b) Cardinal number of the empty set is zero, i.e., n(ϕ) = 0

7) Equivalent Sets

Two finite sets A & B are said to be equivalent, if they have the same number of elements, i.e., if n(A) = n(B)

The equivalence of two sets A & B is expressed by writing A ↔ B

Ex.2. Let A = {a, b, c, d}, B = {p, q, r, s}, then since each set contains 4 elements, i.e.,

n(A) = n(B) = 4, therefore, they are equivalent sets.

For Example -

1) let A set of the letters of the word ‘PAN’, B = set of the letters of the word ‘NAP’, C = set of letters of the word ‘PEN’, then A = B, A ≠ C, B ≠ C

Also, since n(A) = n(B) = n(C), we have  A ↔ B,  A ↔ C,  B ↔ C

It is obvious from the definition of equal and equivalent sets that equal sets are always equivalent but equivalent sets may be or may not be equal.

8) Overlapping Sets

Two sets will be called overlapping sets if, they have at least one element in common

For example, the sets A = {1, 3, 5, 7} and B = {2, 3, 9, 11} are overlapping sets, since the element 3 is common in them.

9) Disjoint Sets -

Two sets will be considered as disjoint set when they have no common elements.

For example, the set R = {1, 3, 5, 7, 9, 11}, S = {2, 4, 6, 8, 10,12} are disjoint sets, since no element is common to them.

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