Question

Write the smallest reflexive relation on set {1, 2, 3, 4}.

Answer 1

Hint: The smallest reflexive relation on set (a, b, c, d) is { (a, a), (b, b), (c, c), (d, d) }. To find the reflexive relation on a given set, take a = 1, b = 2, c = 3 and d = 4 in this question.

We are given a set {1, 2, 3, 4} and we have to find the smallest reflexive relation the given set.
Before proceeding with this question, we will see what a “set” is.
A set in mathematics is a collection of well- defined and distinct objects, considered as an object in its own right. The most basic property is that a set “has” elements.
Like, we are given a set {1, 2, 3, 4} whose elements are 1, 2, 3 and 4.
Now the relations on sets are nothing but the properties of sets. We have basically three types of relations on set - that are,
1) Reflexive relation
2) Transitive relation
3) Symmetric relation
Now, we know that reflexive relation is the relation in which each element of a set is related to itself.
That means if we have set {a, b, c, d} then the smallest reflexive relation on this set is { (a, a), (b, b), (c, c), (d, d) }.
This is the smallest relation because this relation does not contain any other relation on set apart from reflexive relation.
Therefore for set {1, 2, 3, 4},
We have,
a = 1
b = 2
c = 3
d = 4
Therefore, the smallest reflexive relation the given set is { (1,1), (2, 2), (3, 3), (4, 4) }.

Note: Students should keep in mind that a solution on set A is reflexive only when the relation contains each and every element of A related to itself. The relation must not leave any element of set A.
Here we have some examples of reflexive relation that are as follows:-
- “is equal to” (equality)
- “is a subset of” (set inclusion)
- “divides” (divisibility)
- “is greater than or equal to”
- “is less than or equal to”
Now we also have some examples of relations that are irreflexive that are as follows:
- “is not equal to”
- “is coprime to” (for integers > 1, since 1 is coprime to itself)
- “is a proper subset of”
- “is greater than”
- “is less than”