Final answer:
The provided statements about the floor and ceiling functions are true or false depending on their mathematical properties. Statements (a) and (c) are true, while statement (b) is false with a provided counterexample. Statement (d) depends on the value of x, and statement (e) is true based on the floor function definition.
Step-by-step explanation:
The student has presented questions related to the properties of the floor and ceiling functions in mathematics. These are important concepts in real analysis and are used to round numbers to their nearest integers up or down respectively.
- floor(ceil(x)) = ceil(x): This statement is true because the ceiling function maps x to the smallest integer greater than or equal to x, and applying the floor function to an integer yields the integer itself.
- floor(x + y) = floor(x) + floor(y): This statement is false. Counterexample: if x = 0.5 and y = 0.5, then floor(0.5 + 0.5) = floor(1) = 1, but floor(0.5) + floor(0.5) = 0 + 0 = 0.
- ceil(ceil(x/2)/2) = ceil(x/4): This statement is true because the ceiling function is idempotent, meaning that applying it twice has the same effect as applying it once.
- floor(\(\sqrt{ceil(x)}\)) = floor(\(\sqrt{x}\)): This statement can be false, depending on x. For example, if x = 2.25, then \(\sqrt{ceil(2.25)}\) = \(\sqrt{3}\) and floor(\(\sqrt{3}\)) = 1, while floor(\(\sqrt{2.25}\)) = floor(1.5) = 1, thus they are equal. However, for x = 2.1, the statement does not hold: \(\sqrt{ceil(2.1)}\) = \(\sqrt{3}\) thus floor(\(\sqrt{3}\)) = 1, but floor(\(\sqrt{2.1}\)) = 1.44... and the floor is 1, so they are equal in this case. The function floor(\(\sqrt{x}\)) can be discontinuous at some points where x is not a perfect square, unlike floor(\(\sqrt{ceil(x)}\)).
- floor(x) + floor(y) + floor(x + y) \(\leq\) floor(2x) + floor(2y): This statement is true, which can be proven using the definition of the floor function and its properties related to the distribution over addition.