Question

please help the set of integers as
Z−....−3,−2,−1,0,1,2,3....

the greatest integer of x is denoted as
⌊x⌋

is defined to be the largest interger k with
k≤x

discuss with your classmates how
⌊x⌋
may be described as a piecewise defined function

Answer 1

also is
⌊a+b⌋=⌊a⌋+⌊b⌋

always true? what if a or b was an integer? test some values make a conjecture and explain your result

Answer 2

This is the floor function. It would truncate the number down to an integer value ot basically remove the digits after the decimal point. This function would give a constant value between integer values. For example:
f(x)=0 for 0=<x<1
f(x)=1 for 1=<x<2
and so on.
f(x)+f(y) would not always be equal to f(x+y), consider 1.4 and 1.7 as x and y.

However f(x)+f(y)=f(x+y) if x-f(x)+y-f(y)<1. And that applies to integer values.
Let me know if you need more explanation.