Determining the Pairs in the Relation R

Oct 13, 2025 by miodeya.com 10 views

Introduction

In mathematics, a relation is a binary relation between elements of two sets. It is a way to describe the connection or relationship between the elements of two sets. In this article, we will explore the concept of relations, specifically the relation R defined on the set A = {1, 2, 3, 4, 5, 6}. We will determine the pairs in the relation R and examine if R is reflexive.

Defining the Relation R

The relation R is defined as:

R = \{(a, b) \mid a - b \geq 2\}

This means that for any pair (a, b) in R, the difference between a and b must be greater than or equal to 2.

Determining the Pairs in the Relation R

To determine the pairs in the relation R, we need to find all possible pairs (a, b) such that a - b ≥ 2.

For a = 1, the possible values of b are 1, 2, and 3, since 1 - 1 = 0, 1 - 2 = -1, and 1 - 3 = -2 are not greater than or equal to 2. However, 1 - 3 is not greater than or equal to 2, so we exclude it. Therefore, the pairs (1, 1), (1, 2) are not in R, but (1, 3) is not in R either.
For a = 2, the possible values of b are 2, 3, and 4, since 2 - 2 = 0, 2 - 3 = -1, and 2 - 4 = -2 are not greater than or equal to 2. However, 2 - 4 is not greater than or equal to 2, so we exclude it. Therefore, the pairs (2, 2), (2, 3) are not in R, but (2, 4) is not in R either.
For a = 3, the possible values of b are 3, 4, and 5, since 3 - 3 = 0, 3 - 4 = -1, and 3 - 5 = -2 are not greater than or equal to 2. However, 3 - 5 is not greater than or equal to 2, so we exclude it. Therefore, the pairs (3, 3), (3, 4) are not in R, but (3, 5) is not in R either.
For a = 4, the possible values of b are 4, 5, and 6, since 4 - 4 = 0, 4 - 5 = -1, and 4 - 6 = -2 are not greater than or equal to 2. However, 4 - 6 is not greater than or equal to 2, so we exclude it. Therefore, the pairs (4, 4), (4, 5) are not in R, but (4, 6) is not in R either.
For a = 5, the possible values of b are 5 and 6, since 5 - 5 = 0 and 5 - 6 = -1 are not greater than or equal to 2. However, 5 - 6 is not greater than or equal to 2, so we exclude it. Therefore, the pairs (5, 5) and (5, 6) are not in R.
For a = 6, the possible value of b is 6, since 6 - 6 = 0 is not greater than or equal to 2. Therefore, the pair (6, 6) is not in R.

However, we can see that the following pairs are in R:

(3, 1)
(4, 2)
(4, 3)
(5, 3)
(5, 4)
(6, 4)
(6, 5)

Therefore, the pairs in the relation R are:

R = \{(3, 1), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5)\}

Determining if R is Reflexive

A relation R on a set A is reflexive if for every a in A, the pair (a, a) is in R.

In this case, we have:

(1, 1) is not in R
(2, 2) is not in R
(3, 3) is not in R
(4, 4) is not in R
(5, 5) is not in R
(6, 6) is not in R

Since none of the pairs (a, a) are in R, the relation R is not reflexive.

Conclusion

Q: What is a relation in mathematics?

A: A relation in mathematics is a binary relation between elements of two sets. It is a way to describe the connection or relationship between the elements of two sets.

Q: What is the difference between a relation and a function?

A: A relation is a binary relation between elements of two sets, while a function is a special type of relation where each element in the domain is related to exactly one element in the codomain.

Q: What is the difference between a reflexive, symmetric, and transitive relation?

A: A reflexive relation is a relation where each element in the set is related to itself. A symmetric relation is a relation where if (a, b) is in the relation, then (b, a) is also in the relation. A transitive relation is a relation where if (a, b) and (b, c) are in the relation, then (a, c) is also in the relation.

Q: What is the relation R = {(a, b) | a - b ≥ 2} on the set A = {1, 2, 3, 4, 5, 6}?

A: The relation R = {(a, b) | a - b ≥ 2} on the set A = {1, 2, 3, 4, 5, 6} is a binary relation where each element in the set is related to elements that are at least 2 greater than itself. The pairs in the relation R are {(3, 1), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5)}.

Q: Is the relation R = {(a, b) | a - b ≥ 2} on the set A = {1, 2, 3, 4, 5, 6} reflexive?

A: No, the relation R = {(a, b) | a - b ≥ 2} on the set A = {1, 2, 3, 4, 5, 6} is not reflexive because none of the pairs (a, a) are in the relation R.

Q: What is the difference between a relation and an equivalence relation?

A: A relation is a binary relation between elements of two sets, while an equivalence relation is a special type of relation where the relation is reflexive, symmetric, and transitive.

Q: What is the relation R = {(a, b) | a = b} on the set A = {1, 2, 3, 4, 5, 6}?

A: The relation R = {(a, b) | a = b} on the set A = {1, 2, 3, 4, 5, 6} is a binary relation where each element in the set is related to itself. The pairs in the relation R are {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}.

Q: Is the relation R = {(a, b) | a = b} on the set A = {1, 2, 3, 4, 5, 6} reflexive?

A: Yes, the relation R = {(a, b) | a = b} on the set A = {1, 2, 3, 4, 5, 6} is reflexive because all of the pairs (a, a) are in the relation R.

Q: What is the relation R = {(a, b) | a > b} on the set A = {1, 2, 3, 4, 5, 6}?

A: The relation R = {(a, b) | a > b} on the set A = {1, 2, 3, 4, 5, 6} is a binary relation where each element in the set is related to elements that are less than itself. The pairs in the relation R are {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}.

Q: Is the relation R = {(a, b) | a > b} on the set A = {1, 2, 3, 4, 5, 6} reflexive?

A: No, the relation R = {(a, b) | a > b} on the set A = {1, 2, 3, 4, 5, 6} is not reflexive because none of the pairs (a, a) are in the relation R.