Let `R:ZrarrZ` be a relation defined by `R = {(a,b):a,b,inZ, a -b in z)` . Show that
(i) `AAa inZ, (a,a) inR`
(ii) `(a,b) inR implies (b,a)inR`
(iii) `(a,b) inR implies (b,c)inRimplies(a,c)inR`

Let R be a relation from Q to Q defined by R={(a,b): a,b in Q and a-b in Z} . Show that (i) (a,a) in R" for all " a in Q (ii) (a,b) in R implies that (b,a) inR (iii) (a,b) in R and (b,c) in R implies that (a,c) in R

Let R be a relation from N to N defined by R={(a,b): a, b in N and a=b^(2)}. Are the following true? (i) (a,a) in R," for all " a in N (ii) (a,b) in R," implies "(b,a) in R (iii) (a,b) in R, (b,c) in R" implies "(a,c) in R . Justify your answer in each case.

Let R be the relation on Z defined by R= {(a,b): a, b in Z, a-b is an integer}. Find the domain and range of R.

Let R be a relation defined by R = {(a, b) : a ge b }, where a and b are real numbers, then R is

Let R be the equivalence relation on z defined by R = {(a,b):2 "divides" a - b} . Write the equivalence class [0].

Let R be the relation in the set N given by R={(a,b), a=b -2, b gt 6}. Choose the correct answer.

Let A = {1, 2, 3} and R = {(a,b): a,b in A, a divides b and b divides a}. Show that R is an identity relation on A.

Let * be a binary operation on the set R defined by a ** b = (a + b)/2 . Show that * is commulative but not associative.

On Z defined * by a ** b = a -b show that * is a binary operation of Z.

Binary operation * on R - {-1} defined by a * b= (a)/(b+a)