There is a topic in mathematics known as Set Theory which mainly deals with the calculations and studies of sets related to mathematical logic, which in informal language is called the collections of objects.
Into a set, several objects of different kinds can be collected together, however, when it comes to the topic Set theory, it involves objects that are related to one another and are mathematically concerned as well or in other words collected objects relevant to the field of study as a whole. The two German mathematicians, Richard Dedekind and Georg Cantor in the year 1870s, are the ones who initiated this study of set theory.
This set theory particularly deals with Relations and the different types of relations which are considered the most important topics in the study of set theory. The theory includes chapters related to sets, relations, and functions that have a relationship with each other. The term Sets refers to a collection of various elements that are in order and the relations and functions of the terms are used in defining several operations that will be performed on the sets. Relations, in fact give us the connection that two different given sets have. Relations can also state the connections that exist within the given sets.
Definition of Relation
In the subject of mathematics, the term
relation is usually used to define a relationship that exists between two given
sets. While considering a case containing two sets, if there exists a
connection between the two then the relation that exists between them can be well-defined
or well-established. The relation can exist only between the elements belonging
to two different sets that are not empty. Let’s take an example, in the
assembly of a school, all the students have to stand and make a queue on the
basis of their heights in the ascending order. This particular example explains
the relationship that exists between the students and their heights. Hence a set
of ordered pairs is known as a relation.
Fig 1: A mapping showing the relation existing from a given set A to a given set B.
The relation is said to be a subset of A x B.
From the above example the pairs in order and which have a connection with each
other are listed below:
Ø
(1,c)
Ø
(2,n)
Ø
(5,
a) and
Ø
(7,n).
The following notations have been you for the
definition of a set:
1. Domain: The set {1,
2, 5, 7} represents the domain and
2. Range: The set {a, c,
n} is the range.
Relationship between Sets and Relations
The terms set, and relation are interconnected with each other as a relation is used to describe the existing relationship between the two different given sets. In order to check if there exists a connection between the available two sets, relations are used which can describe the relationship. An example for this, two given sets are said to have an empty relation in the case if these two sets do not contain elements which are same.
Domain and Range
The two main words used in set theory are the
Domain and Range of a
relation.
The domain is said to be the set containing all the first elements from the
ordered pairs as discussed in the above section. Range is a term that is used
for a set that contains all the second elements of the given ordered pairs. There
is an easy trick in knowing the range of a relation. In the above mapping we
see that Set B contains elements that are in a relation with the first set.
Hence set B can be either equal to the range of relation or can be bigger.
Let us consider x and y to be the two sets where elements from each form an ordered pair. Let’s say that both set x and set y have a relation, then all the elements belonging to set x are said to be the domain i.e., the set of the first elements of the ordered pairs and the ones under set y, i.e., the set of all the second elements of the ordered pairs are said to be the range. Thus, domain can be written as domain (R) = {a ∈ A: (a, b) ∈ R for some b ∈ B} and range of R = {b ∈ B: (a, b) ∈R for some a ∈ A}.
Hence, Domain (R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}
Inverse Relation
The relation is inverse and becomes observable
only when elements of a particular set have inverse pairs of the other set. Example,
if set A is {(a, b), (c, d)}, the inverse relation is R-1 = {(b, a), (d, c)}.
Therefore, R-1 = {(b, a): (a, b) ∈ R}
Let R be a relation. Consider R to be {(a,8), (1,6), (2,4)}12 then Domain is {a,1,2} and range is {8,6,4}. The inverse of R is {(8,a), (6,1), (4,2)}. The domain then = {8,6,4} and range = {a, 1,2}.
Learn
more: Relations from Class
11 Maths
