Section 23 Consider the function fx ceiling x floor x cel

(Section 2.3) Consider the function f(x) = (ceiling x - (floor x ) celing), with domain R+ and codomain N. (a) Give examples to show that f(x) is neither one-to-one nor onto. (b) Show that f(x) = 0 when x is an integer, and f(x) = 1 when x is not an integer. (Note: I am not asking for a proof here, but be sure that your reasoning is clear enough for me to follow.)

Solution

Here we have the definition that celing(x) is actually the greatest integer function that returns smallest integer among all integers that are greater than given number. And floor (x) is smallest integer function that returns greatest integer among all integers that are smaller than given number.

So for example : ceiling (2.3) = 3 where floor (2.3) = 2

that means f(x) =ceiling x- celing( floor (x ))= 3-ceiling(2) = 3-2=1

Also if x=2.9, even then

ceiling (2.9) = 3 where floor (2.9) = 2

that means ceiling x- celing( floor (x ))= 3-ceiling(2) = 3-2=1

that means for infinite real values between 2 and 3, all have the same range i.e. 1.

That means it is not ONE-ONE as well as it is not ONTO also, because only value 1 codomain N is associated with values of domain x and other natural numbers will not be the image of any domain value.

It proves the part (a).

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Now we just proved in above example , that when x is not an integer, then surely f(x) = 1

and if we let that x= 4 ( an integer), we find that

ceiling (4) = 4 where floor (4) = 4

that means f(x) = ceiling x- celing( floor (x ))= 4-ceiling(4) = 0

So clearly f(x) = 0 when x is integer.

This is the answer of part (b)