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A relation in mathematics represents the interlink between two different sets. For a given pair of sets, the relation between them will be derived, if there is a connection between the elements of these two sets.
When there is no relation between any elements of a set that scenario is called empty or void relation.
Example, If set A={1, 2, 3} then one of the void relations can be R={p, q} where,|p – q| =8
Let’s take another example to get deep insight of the concept:
For given set A = {all the students in a school}
R = {(a,b): Difference between age of a and b is more than 150 years}
Now, we know that no two students in the school will have an age difference of more than 250 years. Even if we take a student of class 1 and take one student of class 12, the age difference will not be more than 150 years. Therefore, None of the elements of this set holds true for the given relation. Hence, it will be an empty relation set.
The relation of any given two sets is the subset of the cartesian product of the sets. Relation used as a tool to identify any connection between a given pair of sets. There are different types of relations among sets such as universal relation, empty or void relation, reflexive relation, symmetric relation etc. Let us discuss the void or empty relation here in this article.
Definition of Empty or Void Relation
The definition of void or empty relation is that if P be a set, then ϕ ⊆P× P and so it is a relation on P. This presentation is called void relation or empty relation on P. In simple words, a relation R on set P is called an empty or void relation, if no element of P is related to any other element of P.
Mathematical Notation :
Mathematically we can write the same as.
R = φ ⊂ P × P
A relation is called an empty relation if none of the elements of any set is mapped with another set or itself. Therefore, it is also referred to as void relation.
No element of set P is mapped with another set Q or set P itself. The empty relation is shown by R=∅.
Void and universal relations are also called trivial relations.
Let’s understand through an elaborated scenario the identification of empty or void relation
Let P be a set, then Φ⊆P X P and so it is a relation on P. This relation is called the void-relation or empty relation on set P.
In simple words, a relation R on set P is called an empty relation, if no element of P is related to any element of P.
For example: The relation R on the set P = {1,2,3,4} defined by
R = {(a,b):a + b = 11}
We observe that a + b ≠ 11 for any two elements of set P . Therefore,
(a,b) ∉ R for any a, b ∈P.
⇒ R does not contain any element P X P
⇒ R is an empty set.
⇒ R is the void relation on P.
Let’s observe another example to understand the empty or void relation concept.
Consider set P consisting of 10 oranges in the basket. Then finding the relation R of getting grapes from the basket is not possible. Since this basket has only oranges and not grapes. Therefore, the above scenario is known as empty or void relation.
Let’s visit a real life scenario to quickly understand the empty or void relation in that context:
Set Q contains the bag of yellow balloons. Then the relation R of getting white balloon from the bag of yellow balloons is what type of relation?
Given set Q contains the bag with the yellow balloons. From the bag of yellow balloons, relation R of getting the white balloon is not possible.Therefore, the above can be described as an empty relation.
The above illustrations help in understanding the concept and tactics of empty or void relation for any given pair of sets.
Conclusion
In this article, we have studied the concept of relation and its definition between two sets. We have also touched the concept of cartesian products with demonstrated examples. We also discussed the meaning of empty or void relation and the methodology to identify the existence of empty or void relation between sets, along with the solved examples. We also discussed daily life scenarios to identify void relation for any given function.
