Q.3 Prove that the Greatest Integer Function f : R\longrightarrow R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Answers (1)

 f : R\longrightarrow R

 f (x) = [x]

One- one:

For   1.5,1.7 \in R   then f(1.5)=\left [ 1.5 \right ] = 1     and    f(1.7)=\left [ 1.7 \right ] = 1

                                 but   1.5\neq 1.7 

\therefore  f is not one- one i.e. not injective.

For  0.6 \in R  there is no x in R such that f(x)=\left [ 0.6 \right ]

\therefore  f is not onto i.e. not surjective.

Hence, f is not injective but not surjective.

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