DHARM4 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCEl If X ⊆ Y and Y ⊆ Z then certainly, X ⊆ Z (transitive).
l A set with no element (empty set) is the subset of all the sets i.e. { } ⊆ X (for any set X)
l {a, b, c} is not a subset of {{a, b, c}}; because former set has the elements a, b, and c but
the latter set has a element which is themselves a set {a, b, c}.
Hence, two sets X and Y are said to be equal if and only if (1) X is the subset of Y and also
(2) Y is a subset of X. Alternatively, if X and Y are two finite sets and if there exists a bijection
between them, then | X | = | Y | is called rule of equality.
For example, {a, b, c} = {a, a, b, c} = {a, c, b} [∴ {a, a, b, c} = {a, b, c} and so |{a, b, c}| =
3 = |{a, c, b}|]. But {a, b, c} ≠ {{a, b}, c}. Also {1} ≠ {{1}} because {1}∈ {{1}} but {1} is not a subset
of the set {{1}}.
Rule of equality of sets is reflexive, symmetric, and transitive that is describes respectively
as,
l A set X is equal to itself i.e., A = A.
l For two sets X and Y if X = Y then also Y = X.
l For any sets X, Y, and Z if X = Y and Y = Z then also X = Z.
After describing the meaning of a subset we shall now define proper subset. Given two
sets X and Y if X ⊆ Y and X is not equal to Y then, set X is called the proper subset of Y and is
denoted by X ⊂ Y. It means, the set Y contains at least one distinct and extra element than the
set X. Therefore, proper subset is not reflexive and symmetric but it is transitive i.e., if X ⊂ Y
and Y ⊂ Z then X ⊂ Z.
Empty Set
A set with no elements is called an empty set. An empty set is also known as a null set and it
is denoted by { } or ∅. For example,
l ∅ = {x/x is an integer and x^2 + 5 = 0}
l ∅ = {x/x are living beings who never die}
l ∅ = {x/x is the MCA student of age below 15}
l ∅ = { x/x is the set of persons of age over 200}
l ∅ = {x/x + 5 = 5 and x > 5}
Remember a set {∅} is not an empty set, because it contains an element ∅. Similarly the
set {{ }} ≠ { }, because former set is not empty, it contains an element ∅ but latter is an empty
set.
1.3.2 Study of Sets Combinations...................................................................................
The sets may be combined into a variety of ways so that new sets are formed. For example,
there is a set of student of girls which is combined with the other set of student of boys then we
get a set of student of either girls, or boys or both girls and boys. What is the set of the senior
students (both girls and boys)? To study these representations ‘union’ and ‘intersection’ are
the basic operations over which the sets are combined. In this section we shall discuss these
sets operations in detail.
Union
Given two sets X and Y, then X union Y denoted by X ∪ Y is the set of all elements either (1)
from the set X, or (2) from the set Y, or (3) from both the set X as well as Y.