question_answer 18)

Check the injectivity and surjectivity of the following functions :       (i) f : F  N given by f(x) = x2       (ii) f : Z  Z given by f(x) = x2       (iii) f : R  R given by f(x) = x2 (iv)  f : N  N given by f(x) = x3 (v)  f : Z  Z given by f(x) = x3  

Answer:

(i) f : N N  given by f(x) = x2 Injectivity : Let x1, x2  such that f(x1) = f(x2)                                      is one ? one. Surjectivity : Let y = 5 be any element              for y = 5 there is no elemnt in N Hence f : N  N is not onto. (ii) f : Z  Z given by f(x) = x2 Injectivity : Let x1 = 2 and x2 = ? 2 be two element to Z.  f(x1) = (2)2 = 4; f(x2) = (?2)2 = 4   f (x1) = f(x2) but x1  x2  f : Z  Z is not one one. Subjectivity : Let y = 5  Z be any element          f(x) = 5        x2 = 5  x =        for y = 5 there is no element in Z       Hence  is not onto.       (iii) f :  given by f(x) = x2       Injectivity : Let x1 = 2 and x2 = ?2 be two element of R.       f(x1) =(2)2 = 4; f(x2) = (?2)2 = 4              is not one-one.       Surjectivity : Let y = ? 5  be any element                    y = ?5, there is no element in R.       Hence f :  is not onto.       (iv) f : N  N given by f(x) = x3       Injectivity : Let x1, x2  such that             f(x1) = f(x2)              is one-one.       Surjectivity : Let y = 4  N be any element                   Hence f :  given by f(x) = x3       Injectivity :Let x1, x2  such that             f(x1) = f(x2)              is one-one. Surjectivity : Let be any element                             is not onto.