Page 76 - ICSE Math 8

Singleton (or unit) set
A set which contains only one (single) element is called a singleton (or unit) set.
For example, if K = {x : x is day of the week starting with the letter F} and L = {x : x is an even prime number},
then in roster form, K = {Friday} and L = {2}. So, K and L are singleton sets.

Empty set (or null set)
A set which does not contain any element is called an empty (or null) set. Maths Info
It is denoted by { } or φ. For example, let M = {x : x is a month of a year • {0} is not an empty set as it
starting with the letter C} and N = {x : x is a prime number between 7 and contains the element ‘0’.
11}. As there is no month which starts with the letter C, M = φ. Similarly, • Similarly, {φ} is also not an
there is no prime number between 7 and 11 so, N = φ. empty set.

Disjoint sets
Two sets are said to be disjoint if they do not have any element in common. For example, if
A = {s, t, u, v} and B = {3, 6, 9, 12, 14}, then A and B are disjoint sets.

Overlapping sets
Two sets are said to be overlapping if they have at least one element in common. For example, if
A = {a, b, c, d} and B = {a, e, i}, then A and B have a as common element. So, they are overlapping sets.

Cardinal Number of a Set Maths Info
The number of elements present in a finite set is called the cardinal number of
that set. The cardinal number of a set say X, is denoted by n(X). For example, Cardinal number of an empty
if A = {a, e, i, o, u} then, number of elements in A = n(A) = 5. set is zero and cardinal number
\ Cardinal number of set A = 5. of an infinite set is not defined.
Similarly, if B = {2, 4, 6, 8, 10, 12, 14, 16}, then n(B) = 8. So, cardinal number of set B = 8.
Example 5: Write the cardinal number of the following sets.
(a) C = {p, q, r, s} (b) D = {2, 3, 5, 7, 11, 13}
Solution: (a) n(C) = 4 \ Cardinal number of set C = 4
(b) n(D) = 6 \ Cardinal number of set D = 6

Equal sets
Two sets are said to be equal if they have the same elements. If sets X and Y are equal, then we write
set X = set Y or X = Y (read as X is equal to Y). For example, if A = {a, b, c, d} and B = {b, d, a, c}, then A = B.

Equivalent sets
Two sets are said to be equivalent if the number of elements is equal in both Maths Info
the sets. The elements can be same or different in the sets. If sets X and Y are All equal sets are equivalent
equivalent, then we write set X ↔ set Y or X ↔ Y (read as X is equivalent sets but all equivalent sets need
to Y). For example, if C = {p, q, r, s, t} and D = {2, 3, 5, 8, 9}, then C ↔ D. not be equal sets.

Subset of a set
Let A and B be any two sets. Set A is said to be a subset of set B if every Try These
element of A is contained (or present) in B. Subset is denoted by ‘⊆’ and if
A is a subset of B, then we denote it by A ⊆ B. Set A = {KKR, DD, RCB, MI}
For example, let A = {2, 3, 5} and B = {1, 2, 3, 4, 5, 6}. In this case, all the Set B = {CSK, SRH, KXIP, RR}
elements of set A, i.e., 2, 3, 5 are also present in set B. So A is a subset of Are these sets equal or
B or A ⊆ B. If A is a subset of B, then B is called the superset of A and we equivalent?
write it as B ⊇ A (read as B is a superset of A).

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