Does The least integer function [-X] = - Greatest integer function -[x]....Give reason of Answers?

Answer 1

i dont know if there is a definition yet of the least integer function... btw lets call it LIF(x)

so let us define it here first... just for fun... lol

LIF(x) = least integer greater than or equal to x.
Note:
GIF(x) = greatest integer less than or equal to x.

Thus LIF(3) = 3... GIF(3) = 3
LIF(4.15) = 5 ... GIF(4.15) = 4
LIF(-3.12) = -3 ... GIF(-3.12) = -4

§
As an additional note:
If GIF(x) = N , N is an integer, then N ≤ x < (N + 1)

similarly
If LIF(x) = N , N is an integer, then N - 1 < x ≤ N


Now to relate the equation:
LIF(-x) = - GIF(x) (? not yet proven)

LIF(-x) = d ... then d - 1 < -x ≤ d
-d ≤ x < -(d - 1) ... -d ≤ x < (1 - d)

d = -GIF(x) then GIF(x) = -d
-d ≤ x < (-d + 1)

thus the two sides are the same.

Answer 2

[-7/2] = -4, but -[7/2] = -3.