Solution Suppose f x f x Note that if x is odd and x is even then we will have x 1 x 1 i e x x 2 which is impossible Similarly the possibility of x being even and x being odd can also be ruled out using the similar argument Therefore both x and x must be either odd or even Suppose both x and x are odd Then f x f x x 1 x 1 x x Similarly if both x and x are even then also f x f x x 1 x 1 x x Thus fis one one Also any odd number 2r 1 in the co domain N is the image of 2r 2 in the domain N and any even number 2r in the co domain N is the image of 2r 1 in the domain N Thus fis onto Example 13 Show that an onto function f 1 2 3 1 2 3 is always one one Solution Suppose fis not one one Then there exists two elements say 1 and 2 in the domain whose image in the co domain is same Also the image of 3 under f can be only one element Therefore the range set can have at the most two elements of the co domain 1 2 3 showing that f is not onto a contradiction Hence fmust be one one Example 14 Show that a one one function f 1 2 3 1 2 3 must be onto Solution Since fis one one three elements of 1 2 3 must be taken to 3 different elements of the co domain 1 2 3 under f Hence fhas to be onto Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary finite set X i e a one one function f X X is necessarily onto and an onto map f X X is necessarily one one for every finite set X In contrast to this Examples 8 and 10 show that for an infinite set this may not be true In fact this is a characteristic difference between a finite and an infinite set EXERCISE 1 2 1 Show that the function f R R defined by f x iii f R iv f N N x where R is the set of all non zero real numbers Is the result true if the domain R is replaced by N with co domain being same as R 1 2 Check the injectivity and surjectivity of the following functions i f N N given by f x x ii f Z Z given by f x x Rationalised 2023 24 is one one and onto R given by f x x given by f x x v f Z Z given by f x x 3 Prove that the Greatest Integer Function f R R given by f x x is neither one one nor onto where x denotes the greatest integer less than or equal to x 1 if x 0 f x 0 if x 0 1 if x 0 RELATIONS AND FUNCTIONS 11 4 Show that the Modulus Functionf R R given by f x x is neither one one nor onto where x is x if x is positive or 0 and x is x if x is negative 5 Show that the Signum Function f R R given by is neither one one nor onto 6 Let A 1 2 3 B 4 5 6 7 and let f 1 4 2 5 3 6 be a function from A to B Show that fis one one