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

Lori, rewrite the problem so that the divisor is a whole number...8.5/2.3

rewrite the problem so that the divisor is a whole number...8.5/2.3

Applications of derivatives, Applications of derivatives : At last, let's ...

Applications of derivatives : At last, let's not forget about our applications of derivatives. Example    Assume that the amount of air in a balloon at any time t is specified

Probability, A card is chosen at random from a pack of playing cards.what i...

A card is chosen at random from a pack of playing cards.what is d probability that it is either a heart or the queen of spades

Calculus three, i would like answers to these questions i will give you as ...

i would like answers to these questions i will give you as soon as possible

Properties of the indefinite integral, Properties of the Indefinite Integra...

Properties of the Indefinite Integral 1.  ∫ k f ( x ) dx = k ∫ f ( x ) dx where k refer for any number.  Thus, we can factor multiplicative constants out of indefinite integral

Progressions, We will look at three types of progressions called Ar...

We will look at three types of progressions called Arithmetic, Geometric and Harmonic Progression. Before we start looking at the intricacies of these let us unders

Elementary row operations, Anne, Betty and Carol went to their local produc...

Anne, Betty and Carol went to their local produce store to buy some fruit. Anne bought one pound of apples and two pounds of bananas and paid $2.11. Betty bought two pounds of appl

Elimination technique of linear equations, What is the Elimination techniqu...

What is the Elimination technique of Linear Equations?

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