Question

10. [12 Points) Properties of relations Consider the relation R defined on R by «Ry x2 - y2 = x - y (a) Show that R is reflexive. (b) Show that R is symmetric. (c) Show that R is transitive. (d) You have thus verified that R is an equivalence relation. What is the equivalence class of 3? (e) More generally, what is the equivalence class of an element x? Use the listing method. (f) Instead of proving the three properties in (a), (b), (c), you could also have exhibited a function f such that xRy # f(c) = f(y). Find such a function f :R → R.

Answer 1

Define a relation on R by xRy x2 - y2 = x - y (a)R is reflexive: x2-x2 = x - x xRx > Ris reflexive (b)xRy >x -y= x >y? - x2 = =yRx .: R is symmetric (c) let xRy,yRz =xRy = x2 - y2 = x - y, yRz = y2 -z? =y-z adding the above two equations then we get (x2 - y2)+(y2 - z?) = (x - y)+(y-2) =x2-22 = x-2 >xRz .: R is transtive relation (d) R is reflexive,symmetric,transtive hence R is an equivalence relation any equivalence relation can partioned the set into equivalence classes such that any two equivalences are either disjoint or identical equivalence class of 3={x ER:x2 – 32 = x-3} (e) equivalence class of an element x ={y € R.x2 - y2 = x-y} (f) the function f:R → R such that f(x)=x².