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Relation

A connection between the elements of two or more sets is Relation. The sets must be non-empty. A subset of the Cartesian product also forms a relation R. A relation may be represented either by Roster method or by Set-builder method.
Let A and B be two sets such that A = {2, 5, 7, 8, 10, 13} and B = {1, 2, 3, 4, 5}. Then,
R = {(x, y): x = 4y – 3, x ∈ A and y ∈ B} (Set-builder form)
R = {(5, 2), (10, 3), (13, 4)} (Roster form)

Types of Relations ​ | Algebra - Mathematics

Types of Relations or Relationship
Let us study about the various types of relations.

Types of Relations ​ | Algebra - Mathematics

Empty Relation
If no element of set X is related or mapped to any element of X, then the relation R in A is an empty relation, i.e, R = Φ. Think of an example of set A consisting of only 100 hens in a poultry farm. Is there any possibility of finding a relation R of getting any elephant in the farm? No! R is a void or empty relation since there are only 100 hens and no elephant.

Universal Relation
A relation R in a set, say A is a universal relation if each element of A is related to every element of A, i.e., R = A × A. Also called Full relation. Suppose A is a set of all natural numbers and B is a set of all whole numbers. The relation between A and B is universal as every element of A is in set B. Empty relation and Universal relation are sometimes called trivial relation.

Identity Relation
In Identity relation, every element of set A is related to itself only. I = {(a, a), ∈ A}. For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). If we define a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is an identity relation.

Inverse Relation
Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1 = {(b, a): (a, b) ∈ R}. Considering the case of throwing of two dice if R = {(1, 2), (2, 3)}, R-1 = {(2, 1), (3, 2)}. Here, the domain of R is the range of R-1 and vice-versa.

Reflexive Relation
If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.

Symmetric Relation
A relation R on a set A is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.

Transitive Relation
A relation in a set A is transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

Equivalence Relation
A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive. For example, if we throw two dices A & B and note down all the possible outcome.
Define a relation R= {(a, b): a ∈ A, b ∈ B}, we find that {(1, 1), (2, 2), …, (6, 6) ∈ R} (reflexive). If {(a, b) = (1, 2) ∈ R} then, {(b, a) = (2, 1) ∈ R} (symmetry). ). If {(a, b) = (1, 2) ∈ R} and {(b, c) = (2, 3) ∈ R} then {(a, c) = (1, 3) ∈ R} (transitive)

Solved Example for You

Problem: Three friends A, B, and C live near each other at a distance of 5 km from one another. We define a relation R between the distances of their houses. Is R an equivalence relation?
Solution: For an equivalence Relation, R must be reflexive, symmetric and transitive.

  • R is not reflexive as A cannot be 5 km away to itself.
  • The relation, R is symmetric as the distance between A & B is 5 km which is the same as the distance between B & A.
  • R is transitive as the distance between A & B is 5 km, the distance between B & C is 5 km and the distance between A & C is also 5 km.

Therefore, this relation is not equivalent.

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FAQs on Types of Relations ​ - Algebra - Mathematics

1. What is a relation in mathematics?
Ans. In mathematics, a relation is a set of ordered pairs where each pair consists of an object from one set called the domain, and an object from another set called the range. It describes how elements from the domain are related to elements in the range.
2. What are the different types of relations in mathematics?
Ans. There are several types of relations in mathematics, including: - Reflexive relation: A relation where every element in the domain is related to itself. - Symmetric relation: A relation where if an element 'a' is related to an element 'b', then 'b' is also related to 'a'. - Transitive relation: A relation where if 'a' is related to 'b' and 'b' is related to 'c', then 'a' is also related to 'c'. - Equivalence relation: A relation that is reflexive, symmetric, and transitive. - Function: A relation where each element in the domain is related to exactly one element in the range.
3. How can we represent relations mathematically?
Ans. Relations can be represented in different ways mathematically, such as: - Set notation: Using curly braces and commas to list the ordered pairs that make up the relation. - Arrow diagram: Using arrows to connect elements from the domain to elements in the range. - Matrix representation: Using a matrix where the rows represent the elements from the domain and the columns represent the elements from the range.
4. What is the difference between a function and a relation?
Ans. While all functions are relations, not all relations are functions. The main difference between a function and a relation is that a function has a unique output for each input. In other words, each element in the domain is related to exactly one element in the range. On the other hand, a relation can have multiple outputs for a single input.
5. How can we determine if a relation is reflexive, symmetric, or transitive?
Ans. - Reflexive: To determine if a relation is reflexive, we need to check if every element in the domain is related to itself. If every element satisfies this condition, the relation is reflexive. - Symmetric: To determine if a relation is symmetric, we need to check if for every pair (a, b) in the relation, the pair (b, a) is also in the relation. If this condition holds true for all pairs, the relation is symmetric. - Transitive: To determine if a relation is transitive, we need to check if for every three elements a, b, and c in the relation, if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation. If this condition holds true for all triples, the relation is transitive.
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