Disprove the following statements by giving a counterexample

Disprove the following statements, by giving a counterexample. For all real number x and y, [xy] middot [y]. For all real numbers n and ma [nm] = [n] middot [m].

Solution

a) Let us assume x=2.4 and y=3.1

Ceil(x) = 3

Ceil(y) = 4

Ceil(xy) = Ceil(2.4*3.1) = Ceil(7.44) = 8

Hence we get Ceil(xy) is not equal to Ceil(x) * Ceil(y), since Ceil(x) * Ceil(y) is equal to 12

b) Let us assume n=2.8 and m=3.4

Ceil(n) = 3

Ceil(m) = 4

floor(m) = 3

Ceil(2.8*3.4) = Ceil(9.52) = 10

Ceil(n) * floor(m) = 9

Hence both the things are not equal