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Equivalence relation is defined on a set in mathematics as a reflexive, symmetric, and transitive binary relation. A subset of the Cartesian product X and Y is a binary relation over the sets X and Y consisting of components of the form (x, y) such that x ∈ X and y ∈ Y. The 'equal to (=)' relation, which is reflexive, symmetric, and transitive, is a highly frequent and simple example of an equivalence relation.
Also Read: Sequence and Series
Key Terms: Binary Relation, Function, Transitive, Reflexive, Symmetric, Relation, Symmetric, Set, Equivalence relation, Subsets
What is an Equivalence Relation?
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A binary equivalence relation is one that is reflexive, symmetric, and transitive and is defined on a set X. The relation cannot be said an equivalence relation if any of the 3 conditions (reflexive, symmetric, and transitive) are not met. The equivalence relation separates the set into equivalence classes that are distinct. If and only if two elements of the set belong to the very same equivalence class, then they are said to be equivalent. The symbol '~'. is commonly used to represent an equivalence relation.
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Terms related to Equivalence Relation
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Equivalence Class - A subset Y of X is an equivalence class if (a, b) ∈ R for any a, b ∈ Y and a, b cannot be outside of Y. An equivalence class of an is denoted mathematically as [a] = {x ∈ X: (a, x) ∈ R}, which comprises all components of X that are connected to 'a'. The equivalence class for all items of X that are equivalent to each other is the same. To put it another way, all components in the same equivalence class are equivalent to one another.
Partition - A partition of set X is a non-empty set of disjoint subsets of A in which no member of X appears in more than one subset and elements from the same subset are connected to one another. Set X is equivalent to the union of the partition's subsets.
Quotient Set - A quotient set is a collection of all equivalence classes of the equivalence relation X/R = {[a]: a ∈ A}
Definition of Equivalence Relation
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Mathematical relations for real numbers R are a function that is defined on a set Q if and only if an is reflexive, symmetric, and transitive, it is said to be an equivalence relation. They're frequently used to group comparable or equivalent objects together. For all elements x, y, and z, it meets the following condition x, y, z ∈ Q:
Reflexive - The relation R is said to be reflexive, if only if all the elements of a given set are mapped to themselves, such that (a, a) ∈ R is true for all a ∈ Q.
Symmetric - On the set Q, the relation R is said to be asymmetric if (a, b) ∈ R, then (b, a) ∈ R, and a, b ∈ Q.
Transitive - If (a, b) ∈ R and (b, c) ∈ R is transitive relations, then (a, c) ∈ R for any a, b, c ∈ R is said to be a transitive relation.
Proof of Equivalence Relation
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Let's look at an example to see how to verify that a connection is an equivalence relation. If and only if a = b, define a relation R on the set of natural numbers N as (a, b) ∈ R. We'll now demonstrate that R is reflexive, symmetric, and transitive.
- Reflexive Property - Since every natural number is the same as itself, a = a for all a N (a, a) R for all an N. As a result, R is reflexive.
- Symmetric Property - Let (a, b) ∈ R ⇒ a = b ⇒ b = a ⇒ (b, a) ∈ R. R is symmetric because a and b are arbitrary.
- Transitive Property - Let (a, b) ∈ R and (b, c) ∈ R ⇒ a = b and b = c ⇒ a = c ⇒ (a, c) ∈ R for a, b, c ∈ N. R is transitive because a, b, and c are arbitrary.
R is an equivalence relation since it is reflexive, symmetric, and transitive when defined on the set of natural numbers N.
Also Read: Linear Equation in Two Variable
Example of Equivalence Relation
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- Is equal to (=) is an example of an equivalence relation on any set of integers A, as we have a = a, a = b ⇒ b = a, and a = b, b = c ⇒ a = c for all elements a, b, c ∈ A. This means that (=) is transitive, symmetric, and reflexive.
- On the set of Triangle, 'is comparable to (~)' is defined: It's transitive, symmetric, and reflexive.
- 'Has the same birthday' been a condition that is applied to a group of people: It's transitive, symmetric, and reflexive.
- The equivalence relation 'Is congruent to' defined on the set of triangles is reflexive, symmetric, and transitive.
- On the set of integers, 'congruence modulo n (≡)' is defined as follows: It's transitive, symmetric, and reflexive.
- The equivalence relation 'has the same absolute value' defined on the set of real numbers is reflexive, symmetric, and transitive.
Equivalence Relation Connection with Others Relations
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- A reciprocal, a system that can be categorised, and a linear relationship are all examples of incomplete orders.
- Equality is both a complete order and a relationship of equivalence.
- A rigorous incomplete order is always asymmetric, bidirectional and irreflexive.
- The ternary comparability relation is the ternary equivalent of the conventional equivalence relation.
- A reliance relation, also known as a tolerance relation, is an asymmetrical and reciprocal relationship.
- Although the converse is true only in classical mathematics because it is the same as the law of excluded middle, any of the equivalence connection is the inverse of an apartness relationship.
- Every recursive Riemann connection, whether left or right, is also an interval estimate.
- A sequence can be both inductive and bidirectional.
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Proving a Relation is not an Equivalence Relation
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Consider the example of a relationship that isn't an equivalence relation and come up with a counterexample. Define a relation R on an integer set as (a, b) ∈ R if and only if a ≥ b. Look for three things (reflexivity, symmetry, and transitivity):
Reflexivity - It satisfies an a for all a Z since every integer is equal to itself, that is, a = a ∀ a ∈ Z. For any a ∈ Z, this implies (a, a) ∈ R. As a result, R is reflexive.
- Let (a, b) ∈ R ⇒ a ≥ b, for a, b ∈ Z. This is not to say that b ≥ a. For example, 12 ≥ 9 is more than or equal to 12, but 9 is not greater or equal to 12. This indicates that R isn't symmetric.
Because R is not symmetric and therefore not an equivalence relation, do not need to check for transitivity.
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Things to Remember
- A binary equivalence relation is one that is reflexive, symmetric, and transitive and is defined on a set X.
- The equivalence relation separates the set into equivalence classes that are distinct.
- Equivalent elements are those that belong to the same equivalence class.
- Equivalent equations are those that have comparable solutions or bases.
- By subtracting or adding the same number or phrase to both sides of an equation, an equivalent equation is generated.
- By dividing or multiplying each side of an equation by a similar non-zero number, a similar equation is generated.
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Sample Questions
Ques. To prove an equivalence relation which of the following 3 conditions must be necessary. (1 mark)
a) Asymmetric, Reflexive and Transitive
b) Asymmetric, Symmetric, and Irreflexive
c) Transitive, Reflexive, and Symmetric
d) Irreflexive, Antisymmetric and Transitive
Ans. c) Transitive, Reflexive, and Symmetric
Hint: A binary equivalent relation is one that is defined on the set X and is reflexive, symmetric, and transitive. The relation cannot be said as an equivalence relation if any of the 3 conditions (reflexive, symmetric, and transitive) are not met.
Ques. What is the relationship with the smallest equivalence? (1 mark)
Ans. The smallest equivalence relation for each set X is the one that contains all pairs (x, x) for all x ∈ X. In mathematics, equivalence relations defined on a set are binary relations such as reflexive, symmetric and transitive relations.
Ques. Is an Equivalence Relation an Empty Relation? (1 mark)
Ans. An empty relation on an empty set is an equivalence relation; however, because it is not reflexive, an empty relation on a non-empty set is not an equivalence relation.
Ques. Is the relationship depicted in the graph below an equivalence relationship? (1 mark)
Ans. This isn't an equivalency relationship. Because (p, p) is absent from the relationship, it is not reflexive.
Ques. Consider the case when A is a relation on the set R real numbers defined by xAy under the condition that x - y is an integer. Show that A is an equivalence relation on R. (3 marks)
Ans. Reflexive Property - Assume that x is a member of R and that x – x = 0 is an integer. As a result, xAx.
Symmetric Property - Assume that R and xAy own x and y, respectively. In addition, x – y is an integer. As a result, y – x = – (x – y) and (y – x) is an integer. As a result, yAx.
Transitive Property - Assume that R, xAy and yAz are the members of x and y. Moreover, x - y and y - z are also integers. As a result, (x – y) + (y – z) = x – z is likewise an integer, according to the transitive property. As a result, xAz.
As a result, R is an equivalence relation of R.
Ques. What are the equivalence relation 3 different properties? (3 marks)
Ans. The following are the three different properties of an equivalence relation:
- Transitive Property
- Symmetric Property
- Reflexive Property
Ques. Show that R = {(5, 5), (6, 6), (7, 7), (5, 6), (6, 7)} is reflexive but neither symmetric or transitive in the set R {5, 6, 7}. (3 marks)
Ans. It is Reflexive because relation R has elements {(5, 5), (6, 6), (7, 7)}.
Since relation R has (5, 6) but it does not contain (6, 5). Hence it is not symmetric.
Since relation R has (5, 6) and (6, 7), but it does not contain (5, 7). Hence it is not transitive.
Ques. On the set of real numbers, the relation "is less than or equal to," denoted "≤" is NOT an equivalence relation. "≤" is reflexive and transitive but NOT necessarily symmetric for every a, b, or c ∈ R. (3 marks)
Ans. (Reflexivity) Since a = a, a ≤ a must be true.
(Symmetry) If a ≤ b, it does not necessarily follow that b ≤ a. For instance, 5 ≤ 7 is not the same as 7 ≤ 5.
(Transitivity) If a ≤ b and b ≤ c is true, then a ≤ b is true since a ≤ b ≤ c.
Ques. "a R b if a – b is divisible by 5" for a, b ∈ Z defines a relation R on the set Z. Check to see if R is a Z equivalence relation. (3 marks)
Ans. Consider a ∈ Z. Then a – a is also divisible by 5. As a result, R is reflexive and aRa holds for all a in Z.
Hold a, b, Z, and aRb. Then, because a – b is divisible by 5, b – n is also divisible by 5.
As a result, aRb ⇒ bRa, and R is symmetric.
Assume that a, b, c ∈ Z and aRb, bRc are both true. Then both a – b and b – c is 5 divisible.
As a result, a – c = (a – b) + (b – c) divides by 5.
As a result, aRb and bRc =⇒ aRc, and R is transitive.
R is an equivalence relation on Z because it is reflexive, symmetric, and transitive.
Ques. Determine whether or not the relation is reflexive, symmetrical or transitive: R in the set P of human beings in a city at a specific moment given by (5 marks)
a) R = {(a, b): a and b are both work at the same place}
b) R = {(a, b): a is 7 centimetres taller than b}
Ans. Let's solve for R = {(a, b): a and b both work in the same place.
Because a and a will work at the same location, the relation will have values (a, a) and (b, b). As a result, it is reflexive.
If a and b both work at the same place, then b and a will also work there.
This relation R will have values (a, b) and (b, a), indicating that it is also transitive.
If a and b work in the same place, and b and c work in the same place, then a and c work in the same place.
As a result, the relation R will have the values (a, b), (b, c) ad (a, c) and is transitive.
As a result, it's an equivalence relation.
Let's look at case number two: R = (a, b): a is 7 centimetres taller than b, hence a - b=7.
a - a = 0 instead of 7. As a result, the relationship will not have (a, a) and it will not be reflexive.
Because a - b ≠ b - c, if relation R has (a, b), it will not have (b, a), indicating that it is not symmetric.
If a - b= 7 and b - c = 7, a - c must = 14, not 7.
As a result, if a relation has both (a, b) and (b, c) parts, it will not have (a, c), indicating that it is not transitive.
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