What is the Difference Between a Congruence an Equivalence Relation?

What is the Difference Between a Congruence an Equivalence Relation?

In abstract algebra, rather than usual number systems, this advanced algebra deals with abstract algebraic structures like groups, rings, vector spaces, modules, lattices, algebras and fields. A few branches of abstract algebra are commutative algebra, representation theory, elementary number theory, discrete mathematics and homological algebra including study areas of logic and foundations, counting, informal set theory, theory of linear operators, etc.

Algebraic concepts are generalised and defined by using symbols and letters to represent the basic arithmetical operations. In abstract algebra, a relation between two Cartesian sets A and B, is an arbitrary subset of product of the sets and described as R, R  A x B. If ordered pair (a, b) is an element of R, where aAand bB, then a is related to b by R. The set of all elements in the codomain B of R to which some a in the domain A is related can be denoted by R(a), which is the set .

Equivalence relations:

In a set A, if a₁ and a₂are related for a relation R, which is a subset of a Cartesian product A×A, written as a₁~a₂ for some a₁, a₂A and (a₁, a₂) R. For all elements of the set A, any relation RA x A following properties are given:

1. Reflexivity: A relation R , a subset of A x A is reflexive iffor all
2. Symmetry: A relation R , a subset of A x A is symmetric if a₁~a₂ a₂~a₂for all a₁, a₂A
3. Transitivity: A relation R , a subset of A x A is transitive if a₁~a₂ and a₂~a₃ a₁~a₃for all a₁, a₂, a₃ A

For which if R exhibits all three properties, then R is called an equivalence relation on A, otherwise it is a non-equivalence relation. Two elements that are related by an equivalence relation are equivalent under the equivalence relation. a₁ and a₂ are equivalent to each other under the equivalence relation R and written as .

The following relations are all equivalence relations:

1. A set of numbers with “is equal to” relation. Example,
2. Same parameter values across a set of population. Example, Equal income on a set of working people.
3. “is similar to” and “is congruent to” on a set of triangles.
4. “having same absolute value” on a set of real numbers.
5. “is congruent to, modulo n” on a set of integers, etc.

Congruence Relation:

Mathematically, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. In abstract algebra, a congruence relation is an equivalence relation on algebraic structures (groups, fields, vector space, etc.) such that within the structure algebraic relations are expressed with equivalent elements to yield equivalent elements. Examples of congruence relations on different algebraic structures are given below:

1. If P is a group with an * operator, the congruence relation on P is an equivalence relation () on the elements of P satisfying,

p₁ p₂ and p₃ p₄ p₁*p₃  p₂*p₄ for all p₁,p₂,p₃,p₄ P.

1. A ring possesses both addition and multiplication. A congruence relation on a ring satisfies

whenever r₁r₂  and s₁s₂, then r₁+s₁r₂ +s₂  and r₁ s₁r₂s₂

The representative example of a congruence relation is the congruence modulo m on the set of integers. Let m be a positive integer for which integers a and b, if m divides b-a, then a is congruent to b modulo m, written as or . Every integer will be congruent exactly to one of the 0,1,2,3, 4…, m-1 integers modulo m. Mathematically, explaining congruence relation and its compatibility to equivalence relation by the following way:

ab for which congruence modulo, for mℕ x ℕ expanded as a=k₁.m+r and b= k₂.m+r.

Example: If a= 4 and b=10, then 410 4=2.2+0 and 10=5.2+0, then 4 is congruent to 10 modulo 2. Proofs that congruence modulo is an equivalence relation by the following explanation:

Reflexitivity: aa a=k₁.m+r and a= k₂.m+r, of which representation n= k.m+ris unique, then k₁ equals to k₂, and reflective for every aℕ

Symmetry: Showing ab → ba

ab a=k₁.m+r and b= k₂.m+r

 b=k₂.m+r and a= k₁.m+r

 ba

Transitivity: Showing ab and bc then ac for cℕ

ab a=k₁.m+r and b= k₂.m+r

 b=k₂.m+r and a= k₁.m+r

bc a=g₁.m+q and b= g₂.m+q

within set ℕ, then g₁ equals to k₂ and q=r.

For which, a=k₁.m+r and c= g₂.m+s

Therefore ac

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In the field of abstract Algebra, it is then explained that a congruence relation or in simple terms congruence is actually an equivalence relation like for example when we talk about a group, ring, and so on and the relation is compatible with the algebraic structure like in cases of algebraic operations that are performed with equivalent elements, equivalent elements will be yielded.