x (iv) f : N → N given by f ( x ) = x (v) f : Z → Z given by f ( x ) = x . 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 . . Show that the Modulus Function f : R → R , given by f ( x ) = | x |, is neither one- one nor onto, where | x | is x , if x is positive or and | x | is – x , if x is negative.
. Show that the Signum Function f : R → R , given by f x ( ) , , , > < if if if is neither one-one nor onto. . Let A = { , , }, B = { , , , } and let f = {( , ), ( , ), ( , )} be a function from A to B.
Show that f is one-one. . In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f ( x ) = – x (ii) f : R → R defined by f ( x ) = + x . Let A and B be sets. Show that f : A × B → B × A such that f ( a , b ) = ( b , a ) is bijective function. .
Let f : N → N be defined by f ( n ) = n n n n + , , if is odd if is even for all n ∈ N . State whether the function f is bijective. Justify your answer. .