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.
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
Expert Answer
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
