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Discover the fundamentals of equivalence relations – a key concept in mathematics. Learn how they define relationships between elements and their important properties.
Contents
What is Equivalence Relation?
An equivalence relation is a fundamental concept in mathematics that defines a particular type of relationship between elements of a set. It’s a binary relation that possesses three key properties: reflexivity, symmetry, and transitivity. Let’s explore these properties in more detail:
Reflexivity: An equivalence relation is reflexive, which means that for every element “a” in the set, “a” is related to itself. In mathematical terms, this property is often expressed as:
Symmetry: An equivalence relation is symmetric, which means that if “a” is related to “b,” then “b” is also related to “a.” Symbolically, it’s expressed as:
Transitivity: An equivalence relation is transitive, which means that if “a” is related to “b” and “b” is related to “c,” then “a” is related to “c.” Mathematically, it’s represented as:
- If a ~ b and b ~ c, then a ~ c
When a relation on a set satisfies all three of these properties—reflexivity, symmetry, and transitivity—it is considered an equivalence relation. Equivalence relations are often used in various branches of mathematics, including set theory, algebra, and topology, as well as in other fields like computer science, where they play a crucial role in defining equivalence classes and partitioning sets into distinct, related subsets.
One common example of an equivalence relation is the “equality” relation on a set of numbers. For instance, in the set of integers, the relation “=” is an equivalence relation because it satisfies the properties of reflexivity (every integer is equal to itself), symmetry (if “a” is equal to “b,” then “b” is equal to “a”), and transitivity (if “a” is equal to “b” and “b” is equal to “c,” then “a” is equal to “c”).
What is an Example of Equivalence Relation?
An equivalence relation is a relation that satisfies three properties: reflexivity, symmetry, and transitivity. Here’s an example of an equivalence relation:
Consider a set of people, and define a relation “is the same age as” on this set. In other words, for any two people A and B, we say that A is the same age as B if and only if they have the same age.
Reflexivity: Every person is the same age as themselves. So, for any person A, A is the same age as A.
Symmetry: If person A is the same age as person B, then person B is also the same age as person A. So, if A is the same age as B, then B is the same age as A.
Transitivity: If person A is the same age as person B, and person B is the same age as person C, then person A is the same age as person C. So, if A is the same age as B, and B is the same age as C, then A is the same age as C.
The relation “is the same age as” on the set of people satisfies all three properties, making it an equivalence relation. Equivalence relations are commonly used in mathematics, particularly in areas like set theory, group theory, and modular arithmetic, as well as in various other fields to establish relationships between objects or elements that share certain properties.
Proof of Equivalence Relation
To prove that a relation on a set is an equivalence relation, you need to show that it satisfies three properties: reflexivity, symmetry, and transitivity. Let’s denote the relation as R on a set A. Here’s how you can prove each property:
Reflexivity:
You need to show that for every element x in A, (x, x) belongs to R.
In other words, for all x ∈ A, (x, x) ∈ R.
This means that every element is related to itself.
Symmetry:
You need to show that if (x, y) belongs to R, then (y, x) also belongs to R.
In other words, for all x, y ∈ A, if (x, y) ∈ R, then (y, x) ∈ R.
This means that if x is related to y, then y is related to x.
Transitivity:
You need to show that if (x, y) belongs to R and (y, z) belongs to R, then (x, z) also belongs to R.
In other words, for all x, y, z ∈ A, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.
This means that if x is related to y and y is related to z, then x is related to z.
If you can prove all three properties for the given relation R, then you have shown that R is an equivalence relation on the set A. It’s important to verify all three properties for the relation to be considered an equivalence relation.
Here’s a simple example to illustrate these properties:
Let R be the relation on the set of integers where (x, y) is in R if and only if x and y have the same parity (both even or both odd).
Reflexivity:
For any integer x, x has the same parity as itself, so (x, x) is in R. Reflexivity is satisfied.
Symmetry:
If (x, y) is in R because x and y have the same parity, then (y, x) is also in R because y and x have the same parity. Symmetry is satisfied.
Transitivity:
If (x, y) and (y, z) are in R, it means that x, y, and z all have the same parity.
Since y has the same parity as both x and z, (x, z) is also in R. Transitivity is satisfied.
Therefore, the relation R on the set of integers is an equivalence relation because it satisfies all three properties.
Properties of Equivalence Relation
An equivalence relation is a fundamental concept in mathematics that describes a particular type of relation between elements in a set. An equivalence relation is characterized by three key properties:
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Reflexivity: For all elements a in the set S, a is related to itself. Mathematically, this is expressed as: a ~ a, for all a in S.
In words, this means that every element in the set is related to itself.
-
Symmetry: If elements a and b in the set S are related, then elements b and a must also be related. Mathematically, this is expressed as: If a ~ b, then b ~ a, for all a, b in S.
In words, this means that the relation is symmetrical, and if two elements are related in one direction, they are also related in the reverse direction.
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Transitivity: If elements a and b are related, and elements b and c are related, then elements a and c must also be related. Mathematically, this is expressed as: If a ~ b and b ~ c, then a ~ c, for all a, b, c in S.
In words, this means that the relation is transitive, and if two elements are related in two successive steps, then they are also related directly.
Equivalence relations are commonly used in various areas of mathematics and other disciplines. They provide a way to partition a set into disjoint subsets (equivalence classes) where elements within the same class are related to each other and not related to elements in other classes. Equivalence relations are used in group theory, graph theory, set theory, and various other mathematical and computer science contexts. They are also fundamental in defining equivalence classes, which can help simplify complex problems by grouping similar elements together.
Equivalence Relation Examples and Solutions
An equivalence relation is a relation that is reflexive, symmetric, and transitive. In other words, for a relation to be an equivalence relation, it must satisfy the following three properties:
Reflexive: For all elements in the set, a ~ a (where ~ is the relation).
Symmetric: If a ~ b, then b ~ a for all elements a and b in the set.
Transitive: If a ~ b and b ~ c, then a ~ c for all elements a, b, and c in the set.
Here are some examples of equivalence relations along with their solutions:
1. Equality (in a set of numbers):
- Relation: For real numbers a and b, a ~ b if and only if a = b.
- Reflexive: a ~ a because any number is equal to itself.
- Symmetric: If a = b, then b = a, so it’s symmetric.
- Transitive: If a = b and b = c, then a = c, so it’s transitive.
2. Congruence (in the set of integers modulo n):
- Relation: For integers a, b, and a positive integer n, a ~ b if and only if a ≡ b (mod n).
- Reflexive: a ≡ a (mod n) because any integer is congruent to itself modulo n.
- Symmetric: If a ≡ b (mod n), then b ≡ a (mod n) because the difference a – b is a multiple of n.
- Transitive: If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n) because the sum of the differences (a – b) and (b – c) is also a multiple of n.
3. Partition of a set:
- Relation: Consider a set S, and let R be a relation on S such that for two elements a and b in S, a ~ b if and only if they belong to the same subset of a partition of S.
- Reflexive: Every element belongs to the same subset as itself, so it’s reflexive.
- Symmetric: If a ~ b, then b ~ a because if a and b belong to the same subset, b and a also belong to the same subset.
- Transitive: If a ~ b and b ~ c, then a ~ c because if a, b, and c belong to the same subset, then a and c also belong to the same subset.
4. Equivalence class in set theory:
- Relation: Given a set S and an equivalence relation ~ on S, the equivalence class of an element a in S is the set {x ∈ S | x ~ a}.
- Reflexive: Every element is related to itself, so it’s reflexive.
- Symmetric: If a ~ b, then b ~ a by symmetry of the equivalence relation.
- Transitive: If a ~ b and b ~ c, then a ~ c by transitivity of the equivalence relation.
These are some examples of equivalence relations and how they satisfy the properties of reflexivity, symmetry, and transitivity. Equivalence relations are used in various mathematical and computational contexts, including partitioning sets, defining equivalence classes, and solving problems in modular arithmetic.
Some Solved Examples on Equivalence Relation
Here are some solved examples of equivalence relations:
Example 1: Equivalence Relation on Integers
Let’s define an equivalence relation on the set of integers, denoted by ℤ, as follows:
For integers a and b, we say that “a is equivalent to b (a ~ b)” if and only if a – b is divisible by 5.
Reflexivity: For any integer a, a – a = 0, which is divisible by 5. Therefore, a ~ a, and the relation is reflexive.
Symmetry: If a ~ b, then a – b is divisible by 5. This implies that b – a is also divisible by 5 (since if x is divisible by 5, then -x is also divisible by 5). So, b ~ a, and the relation is symmetric.
Transitivity: If a ~ b and b ~ c, then a – b and b – c are both divisible by 5. Therefore, (a – b) + (b – c) = a – c is also divisible by 5. Hence, a ~ c, and the relation is transitive.
Therefore, the relation “a is equivalent to b if a – b is divisible by 5” is an equivalence relation on the set of integers ℤ.
Example 2: Equivalence Relation on a Set of People
Consider a set S of people, and define an equivalence relation R on S as follows:
For any two people x and y in S, x R y if and only if they were born in the same year.
Reflexivity: Every person is born in the same year as themselves, so x R x for all x in S. Hence, the relation is reflexive.
Symmetry: If x R y, it means they were born in the same year. Therefore, y R x as well, and the relation is symmetric.
Transitivity: If x R y and y R z, it means x, y, and z were all born in the same year. Consequently, x R z, and the relation is transitive.
Thus, the relation “x is born in the same year as y” is an equivalence relation on the set of people.
Example 3: Equivalence Relation on a Set of Colors
Let C be a set of colors, and define an equivalence relation E on C as follows:
For any two colors c1 and c2 in C, c1 E c2 if and only if they are shades of the same primary color (e.g., different shades of red, blue, or green).
Reflexivity: Every color is a shade of itself, so c1 E c1 for all c1 in C. Hence, the relation is reflexive.
Symmetry: If c1 E c2, it means they are shades of the same primary color. Therefore, c2 E c1 as well, and the relation is symmetric.
Transitivity: If c1 E c2 and c2 E c3, it means c1, c2, and c3 are all shades of the same primary color. Consequently, c1 E c3, and the relation is transitive.
Therefore, the relation “c1 is a shade of the same primary color as c2” is an equivalence relation on the set of colors.
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