Final answer:
The proof for the statements ceil(-x) = -floor(x) and floor(-x) = -ceil(x) is based on understanding the definitions of floor and ceiling functions and applying the properties of inequalities when multiplied by a negative number.
Step-by-step explanation:
To prove the statements ceil(-x) = -floor(x) and floor(-x) = -ceil(x), we need to understand the definitions of floor and ceiling functions. The floor function of a real number x, denoted as floor(x), is the greatest integer less than or equal to x, while the ceiling function, denoted as ceil(x), is the least integer greater than or equal to x.
Let's consider a real number x. Suppose n is an integer such that n ≤ x < n+1. By the definition of the floor function, we have floor(x) = n, and by the definition of the ceiling function, we have ceil(x) = n+1 if x is not an integer.
Now, consider -x. Because n ≤ x < n+1, multiplying these inequalities by -1 (and remembering that multiplying by a negative number reverses the inequalities), we get -(n+1) < -x ≤ -n. This implies that floor(-x) = -(n+1) and ceil(-x) = -n, since -(n+1) is the greatest integer less than or equal to -x, and -n is the least integer greater than or equal to -x. Finally, we can conclude the following:
floor(-x) = -(n+1) = -ceil(x)
ceil(-x) = -n = -floor(x)
Thus, we have proven that ceil(-x) = -floor(x) and floor(-x) = -ceil(x) for any real number x.