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Mathematics for Biotechnology and Data Science
Chapter 1: Set Theory
Chapter 2: Operation of Sets
Chapter 3: Application of Set Theory
Chapter 4: Venn Diagram
Chapter 5: Relation
Chapter 6: MAPPING OR FUNCTIONS
Chapter 8: Inverse of a Function
Relations
1.
Introduction
: Relations:
As we know and already studied
about notions of relations and functions, domain, co-domain, and range, along
with different types of specific real-valued functions and their graphs. The
concept of the term ‘relation’ in mathematics has been drawn from the meaning
of relation in the English language, according to which two objects or
quantities are related if there is a recognisable connection or link between
the two objects or quantities. Let A be the set of students of Class XII of a
school, and B be the set of students of Class XI of the same school. Then some
of the examples of relations from A to B are
(1) 
(2) 
(3) 
(4) 
(5) 
However,
abstracting from this, we define mathematically a relation R from A to B as an
arbitrary subset of 
.
If
(a, b) ∈ R, we say that a is
related to b under the relation 
and we write as a b. In general, (a, b) ∈ R, we do not bother
whether there is a recognisable connection or link between a and b. As we
already know the functions are special kind of relations.
In
this chapter, we will study different types of relations and functions,
composition of functions, invertible functions and binary operations.
2.
Types
of Relations
In
this section, we would like to study different types of relations. We know that
a relation in a set A is a subset of 
. Thus, the empty set 
and are two extreme relations. For illustration,
consider a relation R in the set A = {1, 2, 3, 4} given by 
. This is the empty set,
as no pair (a, b) satisfies the condition a – b = 10. Similarly, 
is the whole set , as all pairs (a, b) in satisfy 
. These two extreme
examples lead us to the following definitions.
A. Empty
Relation
A
relation R in a set A is called empty relation, if no element of A is related
to any element of A, i.e., 
.
B. Universal
Relation
A
relation R in a set A is called universal relation, if each element of A is
related to every element of A, i.e., R = A × A.
Both
the empty relation and the universal relation are sometimes called trivial
relations.
Example
1
Let A be the set of all students of a boy’s school. Show that the relation R in
A given by 
is the empty relation and R'
= {(a, b) : the difference between heights of a and b is less than 3 meters}
is the universal relation.
Solution
Since the school is boy’s school, no student of the school can be sister of any
student of the school. Hence, 
, showing that R is the
empty relation. It is also obvious that the difference between heights of any
two students of the school has to be less than 3 meters. This shows that 
is the universal relation.
Example 2: Set A is the set of all
students of a boy’s school. Show that the relation R on P given by
.
Solution It is obvious that the difference
between the heights of any two students of the school has to be less than 5
metres. Therefore, it can be concluded that 
for all 
.
⇒ 
⇒ R is the universal-relation on set P.
Example 3 Let set 
. Let a relation R be defined on A as 
. Show that R is and empty relation.
Solution The Cartesian
product is the set of all possible ordered pairs of elements from set A.
The relation R is
defined by the condition 
. You must check each ordered pair in 
to see if
it satisfies this condition.
For 
, for 
, and for 
Then 
, for 
and for 
Then 
, for 
and for 
none of the ordered
pairs from satisfy the condition , the contain no
elements.
The
relation is an empty relation.
Remark
In earlier class, we have seen two ways of representing a relation, namely
raster method and set builder method. However, a relation R in the set {1, 2,
3, 4} defined by 
is also expressed as 
if and only if 
by many authors. We may also use this
notation, as and when convenient. If 
, we say that a is
related to b and we denote it as .
One
of the most important relation, which plays a significant role in Mathematics,
is an equivalence relation. To study equivalence relation, we first consider
three types of relations, namely reflexive, symmetric and transitive.
C. Equivalence
Relation
A
relation R in a set A is called equivalence relation then the following 3
condition must be satisfied:
a. Reflexive,
if 
, for every 
b. Symmetric,
if 
implies that , for all 
.
c. Transitive,
if and 
implies that 
, for all 
.
Remark
A relation R in a set A is said to be an equivalence relation if R is
reflexive, symmetric and transitive.
Example
4
Let 
be the set of all triangles in a plane with R
a relation in T given by 
. Show that R is an
equivalence relation.
Solution
R is reflexive, since every triangle is congruent to itself. Further, 
is congruent to 
is congruent to 
. Hence, R is symmetric.
Moreover, 
is congruent to T2 and T2 is congruent to 
is congruent to 
. Therefore, R is an
equivalence relation.
Example
5
Let L be the set of all lines in a plane and R be the relation in L defined as 
. Show that R is
symmetric but neither reflexive nor transitive.
Solution
R is not reflexive, as a line 
cannot be perpendicular to itself, i.e., 
. R is symmetric as 
⇒ 
⇒ 
⇒ 
.
R
is not transitive. Indeed, if 
is perpendicular to 
and is perpendicular to 
, then can never be perpendicular to . In fact, is parallel to , i.e., 
.
Example
6
Show that the relation R in the set {1, 2, 3} given by 
is reflexive but neither symmetric nor
transitive.
Solution
R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R. Also, R is not
symmetric, as 
but 
. Similarly, R is not
transitive, as and 
but 
.
Example
7
Show that the relation R in the set Z of integers given by 
is an equivalence relation.
Solution
R is reflexive, as 
for all a ∈
Z. Further, 
, then 
. Therefore, 
. Hence, 
, which shows that R is
symmetric. Similarly, if and 
, then 
and 
are divisible by 2. Now, 
is even. So, 
. This shows that R is
transitive. Thus, R is an equivalence relation in Z.
Example
8
Let R be the relation defined in the set 
by 
. Show that R is an
equivalence relation. Further, show that all the elements of the subset {1, 3,
5, 7} are related to each other and all the elements of the subset {2, 4, 6}
are related to each other, but no element of the subset {1, 3, 5, 7} is related
to any element of the subset {2, 4, 6}.
Solution
Given any element a in A, both a and a must be either odd or even, so that . Further, ⇒
both a and b must be either odd or even ⇒ . Similarly, and ⇒
all elements a, b, c, must be either even or odd simultaneously ⇒ 
. Hence, R is an
equivalence relation. Further, all the elements of {1, 3, 5, 7} are related to
each other, as all the elements of this subset are odd. Similarly, all the
elements of the subset {2, 4, 6} are related to each other, as all of them are
even. Also, no element of the subset {1, 3, 5, 7} can be related to any element
of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements of {2, 4, 6}
are even.
