Question

Prove or disprove:
floor(xy) = floor(x)*floor(y) for all real numbers x and y.

Answer 1

Step-by-step explanation:

The statement "floor(xy) = floor(x) * floor(y) for all real numbers x and y" is not true in general. I'll provide a counterexample to disprove this statement.

Counterexample:

Let's consider x = 1.5 and y = 2.5.

floor(x) = floor(1.5) = 1

floor(y) = floor(2.5) = 2

However, floor(xy) = floor(1.5 * 2.5) = floor(3.75) = 3

On the other hand, floor(x) * floor(y) = 1 * 2 = 2

Since 3 ≠ 2, the equality does not hold for this counterexample, which disproves the statement for all real numbers x and y.

In general, the floor function does not distribute over multiplication for real numbers.