Question

5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer

Answer 1

Solution :

Q1.

(a) f(2) = {-1,0,1}

as y^2 < x and x = 2

(b) f(x) cannot be injective as one value is linked to more than one value or none. Also, f(x) is not surjective as not all the values are covered in f(x) set of values as 2 can be linked to only values below it. that is n can be linked to max n-1, not n itself hence these sets are neither injective nor surjective nor bijective.

Q2.

(a) f is injective

because every number is connected or linked to unique number of f(x) set

let n be even, it is linked to n+1 an odd number

next even number will be n+2 which will be linked to the n+2+1 number which is odd

hence even numbers are linked to every consecutive odd number same is the case in odd number n it is linked to even number n-3.

(b) yes, the set is surjective because two consecutive even numbers are linked to two consecutive odd numbers and no integer is left out in the process.