We define the floor and ceiling of a number x ER as follows: the floor of...

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We define the floor and ceiling of a number x ER as follows: the floor of x, denoted [x], is the unique largest integer that is less than or equal to x. - the ceiling of x, denoted [x], is the unique smallest integer that is larger or equal to x. Hence, the floor and ceiling of a real number are the integers to the immediate left respectively right of that number. If the number is itself an integer, then its floor and ceiling coincide and are equal to the number itself, i.e., for ne Z, we have [n] = [n] = n. (1) Find the floor and ceiling of: i) 25/4 ii) 0.999 izi) -2.01. (2) Is it true that for any real numbers x₁, i = 1, 2, we have [₁ + x₂] = [₁] + [₂]? Justify your answer. (3) Is it true that for any real number x and any integer n we have [x+ n] = [x] + n? Justify your answer. We define the floor and ceiling of a number x ER as follows: the floor of x, denoted [x], is the unique largest integer that is less than or equal to x. - the ceiling of x, denoted [x], is the unique smallest integer that is larger or equal to x. Hence, the floor and ceiling of a real number are the integers to the immediate left respectively right of that number. If the number is itself an integer, then its floor and ceiling coincide and are equal to the number itself, i.e., for ne Z, we have [n] = [n] = n. (1) Find the floor and ceiling of: i) 25/4 ii) 0.999 izi) -2.01. (2) Is it true that for any real numbers x₁, i = 1, 2, we have [₁ + x₂] = [₁] + [₂]? Justify your answer. (3) Is it true that for any real number x and any integer n we have [x+ n] = [x] + n? Justify your answer.

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