LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

TYPES OF SETS

**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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