What is equivalence relation, Mathematics

Assignment Help:

What is equivalence relation?  Prove that relation  'congruence modulo' (  ≡mod m) is an equivalence relation. 

Ans: A relation R illustrated on a nonempty set A is said to be an equivalence relation if R is Reflexive, Symmetric and Transitive on A. 

Any integer x is said to 'congruence modulo m' other integer y, if both x and y yield similar remainder while divided by m. Let R be the relation 'congruence modulo m' over set of integers Z. 

Reflexivity: Let x ∈ Z be any integer, after that x ≡m x since both yield similar remainder while divided by m. Thus, (x, x) ∈ R ∀ x ∈ Z. This proves that R is a reflexive relation. 

Symmetry: Let x and y be any two integers and (x, y) ∈ R. This depicts that x ≡m y and hence y ≡m x. So, (y, x) ∈ R. Therefore R is a symmetric relation also.

Transitivity: Let x, y and z be any three elements of Z like (x, y) and (y, z) ∈ R. So, we have x ≡3y and y ≡mz.  It defines that (x-y) and (y-z) are divisible by m. Hence, (x - y) + (y - z) = (x - z) is as well divisible by m i.e. x ≡m z. Therefore, (x, y) and (y, z) ∈ R ⇒ (x, z) ∈ R. Thus R is a transitive relation.    

Hence, R is an equivalence relation.


Related Discussions:- What is equivalence relation

Write down two more reasons why division is difficult, Write down two more ...

Write down two more reasons why children consider 'division' difficult. Regarding the first reason given above, one of fie few division related experiences that the child perhaps d

Divide 6.8 × 105 by 2.0 × 102 write your answer scientific, Divide 6.8 × 10...

Divide 6.8 × 10 5 by 2.0 × 10 2 . Write your answer in scientific notation? To divide numbers written in scienti?c notation and divide the ?rst numbers (6.8 ÷ 2.0 = 3.4); the

Square and square root., the value of square root of 200multiplied by squar...

the value of square root of 200multiplied by square root of 5+

Triangles, ABCD is a parallelogram which AB and CD are divides by P and Q. ...

ABCD is a parallelogram which AB and CD are divides by P and Q. Such that AP:PB=3:2 and CQ:QD=4:1. If PQ and AC are meet at R, show that AR=3/7AC.

Transportation problems vogel approximation method, if there is a tie betwe...

if there is a tie between two penalties then how to make allocations?

Decision trees and sub sequential decisions, Decision Trees And Sub Sequent...

Decision Trees And Sub Sequential Decisions A decision tree is a graphic diagram of different decision alternatives and the sequence of events like if they were branches of a t

What is the diameter of the pipe along with the insulation, A pipe has a di...

A pipe has a diameter of 2.5 inches. Insulation which is 0.5 inches thick is placed around the pipe. What is the diameter of the pipe along with the insulation around it? The i

Applying quadratics math question, A boat tour company charges $11 for a ha...

A boat tour company charges $11 for a harbour tour and averages 450 passengers on Saturdays. Over the past few months, the company has been experimenting with the price of a tour a

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd