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An equivalence relation is a sort of binary relation that should be reflexive, symmetric plus transitive. In set theory, a relation is defined as a way of showing a connection between any two sets. A relation in mathematics defines the link between two distinct sets of information. If two sets are considered, the relation between them will be confirmed if there is an association between the elements of the given sets.
There are 8 main types of relations which involve: empty relation, identity relation, universal relation, symmetric relation, transitive relation, equivalence relation, inverse relation and reflexive relation.
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Equivalence Relation is a sort of binary relation that should be reflexive, symmetric plus transitive in nature. The well-known instance of an equivalence relation is the “equal to (=)” relation. In other words, we can consider when two elements of the provided set are equivalent to each other if they relate to the same equivalence class.
Condition for equivalence:
Equivalence relation example:
If A = {3, 4, 5}, then relation R = {(3, 3), (4, 4), (5, 5), (3, 5), (5, 3), (3, 4), (4, 5)} is an equivalence relation ∵ the relation R is reflexive, symmetric and transitive as shown above respectively.
In terms of mathematical concepts, a binary relation over the sets A and B denotes a subset of the cartesian product A × B consisting of components of the form (a, b) such that a ∈ A and b ∈ B. If any of the three conditions that are reflexive, symmetric or transitive is not supported, the relation cannot be an equivalence one. An equivalence type of relation is commonly expressed by the symbol ‘~’.
The properties of equivalence relations include reflexive, symmetric, and transitive. These ensure that every element relates to itself, mutual relationships hold in both directions, and consistency is maintained across linked elements. Together, they define a balanced and logical way to group related elements.
The reflexive property means that every element is related to itself.
So, if we take any number a from a set A, the pair (a, a) must be in the relation R.
Now, for positive integers, we can write:
((a, b), (a, b)) ∈ R
This is true because multiplying the same numbers gives the same result:
ab = ab for all positive integers.
So, the reflexive property is satisfied.
The symmetric property says that if one pair is in the relation, then its reverse must also be in the relation.
In other words, if (a, b) ∈ R, then (b, a) must also be in R.
So, for the relation, if ((a, b), (c, d)) ∈ R, then ((c, d), (a, b)) must also belong to R.
Given that ((a, b), (c, d)) ∈ R, it means ad = bc.
And because multiplication is commutative (order doesn’t matter), cb = da.
So, we also have ((c, d), (a, b)) ∈ R.
Hence, the symmetric property is true.
The transitive property means that if the first pair is related to the second, and the second is related to the third, then the first is related to the third.
So if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
For positive integers, if we have:
((a, b), (c, d)) ∈ R and ((c, d), (e, f)) ∈ R,
we must check if ((a, b), (e, f)) ∈ R also holds.
From the given,
ad = cb and cf = de
This means:
a/b = c/d and c/d = e/f,
So, clearly: a/b = e/f,
which gives af = be
Thus, ((a, b), (e, f)) ∈ R.
Therefore, the transitive property is also true.
An equivalence relation is a special rule used to compare elements in a set.
For a relation to be called an equivalence relation, it must follow three rules:
Reflexive (every element is related to itself),
Symmetric (if one element is related to another, then the reverse is also true), and
Transitive (if one is related to a second, and the second to a third, then the first is related to the third).
Equivalence relations group elements into equivalence classes, where all members of the same group are considered equal or related to each other.
These groups don’t overlap, and every element in the set belongs to one and only one equivalence class.
We already know how to prove if a relation is an equivalence relation. Now, let’s look at an example of a relation that is not one.
Consider a relation R on the set of all integers. The rule is: (a, b) is in R if and only if a is greater than or equal to b (written as a ≥ b).
Let’s check the three important properties:
Reflexive:
Every number is equal to itself, so a ≥ a is true for any integer a.
This means the relation is reflexive.
Symmetric:
If a ≥ b, does that mean b ≥ a? Not always.
For example, 12 ≥ 9 is true, but 9 ≥ 12 is false.
So, the relation is not symmetric.
Since the relation is not symmetric, there’s no need to check the third property (transitive). If any one property fails, the relation is not an equivalence relation.
A binary relation on a given set is supposed to be an equivalence relation, if and only if it is reflexive, symmetric and transitive.
i.e. for all p, q, r in set X:
p ∼ p (Reflexivity).
p ∼ q if and only if q ∼ p (Symmetry).
If p∼q and q∼r, then p∼r (Transitivity).
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To learn how to prove if a relation is an equivalence one and determine how many equivalence relations are on a set; let us consider an example.
As P, explained on the set of natural numbers N, is reflexive, symmetric, plus transitive, and P is an equivalence relation.
The equivalence relationships can be explained in terms of the following examples:
Example 1: Prove that the relation R is an equivalence type in the set P= { 3, 4, 5,6 } given by the relation R = { (p, q):|p-q| is even }.
Solution:
R = { (p, q):|p-q| is even }. Where p, q belongs to P.
Reflexive Property :
From the provided relation |p – p| = | 0 |=0.
So R is Reflexive.
Symmetric Property :
From the given relation |p – q| = |q – p|.
Therefore R is symmetric.
Transitive Property :
Accordingly,
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Example 2: Consider A = {2, 3, 4, 5} and R = {(5, 5), (5, 3), (2, 2), (2, 4), (3, 5), (3, 3), (4, 2), (4, 4)}.
Confirm that R is an equivalence type of relation.
Reflexive: Relation R is reflexive because (5, 5), (2, 2), (3, 3) and (4, 4) ∈ R.
Symmetric: Relation R is symmetric as whenever (a, b) ∈ R, (b, a) also relates to R.
Example: (3, 5) ∈ R ⟹ (5, 3) ∈ R.
Transitive: Relation R is transitive as whenever (a, b) and (b, c) relate to R, (a, c) also relates to R.
Example: (3, 5) ∈ R and (5, 3) ∈ R ⟹ (3, 3) ∈ R.
Accordingly, R is reflexive, symmetric and transitive. So, R is an Equivalence Relation.
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Here is a brief summary of the various types of relations in discrete mathematics along with the representation:
Types of Relations |
Representation |
Empty Relation |
R = ∅ ⊂ A × A |
Universal Relation |
R = A × A |
Identity Relation |
I or \(I_{A}\)= {(a, a), a ∈ A} |
Inverse Relation |
\(R^{-1}\) = {(b, a): (a, b) ∈ R} |
Reflexive Relation |
(a, a) ∈ R |
Symmetric Relation |
aRb ⇒ bRa, ∀ a, b ∈ A |
Transitive Relation |
aRb and bRc ⇒ aRc ∀ a, b, c ∈ A |
An equivalence type of relation signifies a binary relation established on a set X such that the relation is reflexive, symmetric and transitive.
Some of the terms associated with an equivalence relation are: Take an equivalence relation R defined on a set A including a, b ∈ A.
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