Thread: Logic Proofs

Recall that the floor of a real number x, i.e. [x], is defined to be the largest integer d such that d <= x

Proposition. For any real numbers x and y, [x + y] >= [x] + [y]

(a) Explain briefly why it's not true if you replace >= with =
(b) Prove that this claim is true.

Notice that the definition of floor folds together two properties. If t is the floor of z, then (i) t <= z and (ii) for any integer d, if d <= z, then t >= d.

It may help to give short names (e.g. like d) to expressions like [x + y].

x could be 1.5 and y could be 1.5 and
[1.5 + 1.5] = [3] = 3 is not equal to [1.5] + [1.5] = 3
I don't know how you work in logic so I can't write a formal proof for you.

Originally Posted by captainjapan

Recall that the floor of a real number x, i.e. [x], is defined to be the largest integer d such that d <= x

Proposition. For any real numbers x and y, [x + y] >= [x] + [y]

(a) Explain briefly why it's not true if you replace >= with =
(b) Prove that this claim is true.

Notice that the definition of floor folds together two properties. If t is the floor of z, then (i) t <= z and (ii) for any integer d, if d <= z, then t >= d.

It may help to give short names (e.g. like d) to expressions like [x + y].

For your part (b), just use the two properties. By property (i), [x] + [y] <= x + [y] <= x + y. Then, since [x] + [y] <= x + y, property (ii) says [x + y] >= [x] + [y].

Could you write a formal proof please.

Thanks

Originally Posted by captainjapan

Could you write a formal proof please.

Thanks

I'm not sure how much more "formal" I can get with what I already said. The only missing part is the statement of assumptions, which you should be able to figure out without cognition (just restate them as they are laid out in the proposition).