Let [a] = greatest integer less than or equal to a (so for example [3.14] = 3, [
ID: 3110616 • Letter: L
Question
Let [a] = greatest integer less than or equal to a (so for example [3.14] = 3, [3] = 3, [-3.14] = -4). This is also called the floor of a. Likewise [a] is the ceiling of a, which is defined as the smallest integer greater than or equal to a (so for example [3.14] = 4, [3] = 3, [-3.14] = -3) a) Consider the function f(x, y) = [x + y] Find the inverse image f^-1 (M) of the set M = {0, 1} in each of the following cases. Support your answer with brief reasoning. (a) Domain N times N, co-domain: Z (b) Domain Z times Z, co-domain: Z (c) Domain R times R, Co-domain: ZExplanation / Answer
a) f-1(M) = {} as sum of any two natural numbers is always greater than or equal to 2. And [n] = n where n is a natural number.
b)f-1(M) = {(x, -x)| x in Z} union {(x, 1-x)| x in Z}. As [x] = x where x is an integer.
c)f-1(M) = { (x,y)| 0 <= x+y <2} as [x] = 0 if x in [0,1) and [x] = 1 if x in [1,2).
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