# Answered Essay: Let R be a binary relation on N^2 defined by: (m, n) R (j, k) iff m + k = n + j. equivalence relation. Prove that R i

Let R be a binary relation on N^2 defined by: (m, n) R (j, k) iff m + k = n + j. equivalence relation. Prove that R is an equivalence relation.

A relation is said to be equivalence if it is reflexive, symmetric and transitive

Reflexive:

A relation is said to be reflexive if aRa is true for the given relation

here for the given relation (m,n)R(m,n) is true because m+n=m+n (since here j=m and k=n) is ture.

hence the relation is reflexive.

Symmetric:

A relation is said to be symmetric if aRb is true the then bRa should be true.

here the relation is (m.n)R(j,k) then (j.k)R(m,n) is true because j+n=k+m

hence the relation is symmetric

Transitive:

A relation is said to be transitive if aRb is true and bRc is true then aRc should be true

consider (a,b)R(c,d) and (c,d)R(e,f) are ture then (a,b)R(e,f) should be true

from (a,b)R(c,d) we get a+d = b+c => a-b=c-d

from (c,d)R(e,f) we get c+f = d+e => c-d=e-f

now for (a,b)R(e,f) a+f=b+e => a-b=e-f

from above equations it is true so the given relation is transitive

Hence the relation is reflexive, symmetric and transitive

hence the relation is equivalence

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