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

Last updated date: 16th Mar 2023

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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”

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”

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