DHARMDISCRETE THEORY, RELATIONS AND FUNCTIONS 3member of the set X. Since, the elements of the set are all unique so incident of repeated
elements doesn’t change the nature of the set.
The important aspect of the study of the sets is its representation such that how can we
describe the members of the sets conveniently. For example, assume a set S contains hundred
million natural numbers. We can represent this set by using the expression that describes its
members conveniently by,
S = {x/x is a natural number and x is upto hundred million}
Consider another example of a set X is the students of UP Technical University study-
ing in MCA program. It can be describe by the following expression by,
X = {x/x is studying in MCA program of UP Technical University}
Here x is the member of the set X and the it’s members share a unique and common
property ‘studying in MCA program’ through which the set X is recognized or by defining x the
set X is completely defined.
Alternatively, a set is well defined if it is possible to determine the members of the set
by means of certain statute.
Since, there is no restriction on the size of the objects that can be in a set. It may possi-
ble that a set contains themselves one or more sets for example,
X = {1, 2, {a, b}, {p, r, s}, $}
Here, sets {a, b} and {p, q, r} are the members of the set X i.e. {a, b} ∈ X and {p, q, r} ∈ X
but the element a, b ∉ X and also p, q, r ∉ X.
The cardinality of the set X is denoted by the | X |, which is the number of elements the
set contains or it also defines the size of the set. | X | may be finite or infinite depending upon
the set finite or infinite. For example, the size of the set X shown above is 5; because it contains
the first two elements 1 and 2, along with two elements that are sets {a, b} and {p, q, r} and one
element $. The cardinality of the empty set is | ∅ | = 0. We set | X | = ∞ whenever set X is
infinite. An infinite set has infinite number of distinct elements. An infinite set that can be put
into a one-to-one correspondence with the natural numbers is countable infinite (i.e. set of
integers) otherwise it is uncountable (i.e. set of reals). The sets have the same cardinality if
their elements shown one-to-one correspondence. A finite set X with | X | = n is called an n-
set. A 1-set is called a singleton.
1.3 SET RULES AND SETS COMBINATIONS
In this section we shall discuss the rules that are the basic tools for enumeration and the
diversity of sets combinations.1.3.1 (S1) Rule of Equality
To discuss the concepts of equality between sets we first discuss the meaning of the subset.
Given two finite sets X and Y if every element of X is an element of Y, then set X is a subset of
Y and it is denoted by, X ⊆ Y. For example, set {a, b, c} is a subset of the set {p, q, r, a, b, w, c};
but not a subset of {p, q, r, a, $, c}. Consider, another example, a set of first year MCA students
is a subset of the set of students that contains all year of MCA students.
Reader should note that,
l A set is subset to itself i.e. X ⊆ X (reflexive).
l If X ⊆ Y then it doesn’t necessarily mean Y ⊆ X.