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A relation in mathematics defines the connection between two different sets. For a given pair of sets, the relation between them will be determined; if there is a relationship between the elements of these two sets.
Consider set A consists of all whole numbers, and set B consists of all integers. Then the relation R:A→B is universal because all the elements of set A are there in set B. (As we know all whole numbers are integers).
The relation R is universal; all the elements of set A are mapped with all the elements of set B or set A itself.
R=A×A or R=A×B.
The subset of the cartesian product of the sets is termed as the relation of two sets.
Relation used as a tool to identify any connection between a given pair of sets. Let us discuss the universal relation in detail below.
Concept of Universal Relation:
Universal relation shows the relation between two sets, if all the elements of the first set are related to the second set, the relation is known as universal.
Note: Second set can be some other set or the first set itself.
Universal relation is represented as relation on set P when P x P⊆P x P. In other words, it is the universal relation if each element of set P is connected to every element of P.
Universal relation is denoted by Ru and is defined on given set A, so here universal relation on given set A as the set form by containing all the ordered pairs that belongs to the cartesian product of A with itself.
Mathematically we can write the same as
Ru (A) = (A x A)
Characteristics of universal relation can be as follows:
1. It is the opposite of the empty or void relation.
2. This relation is defined from and to the same set B.
3. Every element in B is related to all elements in B.
4. It can be denoted as R=A×B.
5. Universal relation shows the largest relation as opposite to the empty relation which shows the smallest possible relation between two sets
6. The combination of empty relation and universal relation is termed as trivial relation
Let’s understand through an elaborated scenario the method of identification of universal relation between sets:
Set A {a1, a2}, the cartesian product of Set A will have below ordered pair.
A x A = {(a1, a1), (a1, a2), (a2, a1), (a2, a2)}
So a universal relation on set A can be defined as a relation in which it consists of all ordered pairs.
Therefore, in mathematical terms Ru (A) = {(a1, a1), (a1, a2), (a2, a1), (a2, a2)}
We can also denote the universal relation using the set-builder form on Set A, consisting of all ordered pair (a,b) where a∈ (belongs to) Set A and also b∈ (belongs to) Set A, this can be expressed as below
Ru (A) = {(a,b): a ∈ A^b ∈ A}
Here is another practical example to understand the universal relation.
Example: Set A is the set of all students of a boy’s school. Show that the relation R on P given by R = {(a,b) : difference between the heights of a and b is less than 5 metres.
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 (a,b) ∈ R for all a, b ∈P.
⇒ R = P X P
⇒ R is the universal-relation on set P.
The above illustrations help in understanding the concept and tactics of universal relation for any given pair of sets.
Conclusion
In this article, we have studied the relation and its definition, which gives the relationship between two sets. We have understood the concept of cartesian products with demonstrated examples. We also discussed the universal relation and the methodology to identify the existence of universal relation between sets, along with the solved examples.
