Help me prove or disprove these claims

[Melodia]

Homework Statement

Heya everyone, I need help proving or disproving these claims:

Homework Equations

The Attempt at a Solution

This definition of the floor totally confused me, I don't know how to start this problem as I don't recognize anything such as axioms or formulas from the claim. Please give me some pointers in the right direction, thanks in advance!

(Yeah I know this is my third question, there are just so many problems and I'm running into a lot of dead ends) ^^

 

[sylas]

... This definition of the floor totally confused me

It may make more sense if you hear the definition read from the formula in English.

The definition you have provided for floor reads as follows: y is equal to the floor of x if and only if

• y is an integer, and
• y is less than or equal to x, and
• for every other integer z, if z is less than or equal to x, then it is also less than or equal to y.

As for the proofs, are you required to use formal proof systems from formal logic in full, or can you be a little bit less formalized?

Cheers -- sylas

 

[Melodia]

Hmm I'm supposed to use the formal method of proving, in this form:

Assume x is in R
     Assume p(x)
          Then r1(x)
          Then r2(x)
          ...
          Then q(x)
     Then p(x) ) q(x). # assuming antecedent leads to consequent
Then for all x in R, p(x) => q(x)

But some general ideas on the steps I should tackle these three problems would be fine too! So I can try to work them out.

 

[wisvuze]

if you can prove it without writing it out "pedantically", then you can probably just translate it over.

for the first question, you can prove that by considering if x is not an integer (if it were an integer, then i) is true by definition). So if x is not an integer, you can express it as a/b, but since a/b is not an integer, this will not divide out evenly. Now you can use the division algorithm and consider the "quotient" and remainder.

 

[sylas]

if you can prove it without writing it out "pedantically", then you can probably just translate it over.

Yes. An informal proof is really just a persuasive outline of how you could make a formal proof if required.

for the first question, you can prove that by considering if x is not an integer (if it were an integer, then i) is true by definition). So if x is not an integer, you can express it as a/b, but since a/b is not an integer, this will not divide out evenly. Now you can use the division algorithm and consider the "quotient" and remainder.

?? Don't use division; division doesn't show up in the problems. I can't see what you are attempting there or how it could work.

For the first problem, you should first sort out for yourself whether it is true or not; this will say whether you want to give a proof or a disproof. For this first step, it can help to look at examples for x and z to help see what is going on. For example, see if it is true for x=1.7 and z=-5.

Floor(1.7) is 1
Floor(1.7 + -5) = Floor(-3.3) = -4 = Floor(1.7) + -5

This looks promising, and might let you think that the theorem is true, so you need a proof.

The proof needs to use the definition you have been given. So can you prove that Floor(x)+z satisfies the required properties to be equal to Floor(x+z)?

Cheers -- sylas

 

[Melodia]

Yes. An informal proof is really just a persuasive outline of how you could make a formal proof if required.



?? Don't use division; division doesn't show up in the problems. I can't see what you are attempting there or how it could work.

For the first problem, you should first sort out for yourself whether it is true or not; this will say whether you want to give a proof or a disproof. For this first step, it can help to look at examples for x and z to help see what is going on. For example, see if it is true for x=1.7 and z=-5.

Floor(1.7) is 1
Floor(1.7 + -5) = Floor(-3.3) = -4 = Floor(1.7) + -5

This looks promising, and might let you think that the theorem is true, so you need a proof.

The proof needs to use the definition you have been given. So can you prove that Floor(x)+z satisfies the required properties to be equal to Floor(x+z)?

Cheers -- sylas

The example makes sense, however I'm now stuck at the actually proving it part ^^
So far I've got:

Assume x is in R and z is in Z
	By definition of floor of x, [x] <= x
		Then ([x] + z) <= (x + z)
		
                ...(but how do I manipulate ([x] + z) <= (x + z) into [x + z] = [x] + z?)
                or do I simply plug in a few values and then if all works, just say it's proven?
	

Then for all x in R and z in Z, [x + z] = [x] + z

 

[Mark44]

The example makes sense, however I'm now stuck at the actually proving it part ^^
So far I've got:

Assume x is in R and z is in Z
	By definition of floor of x, [x] <= x
		Then ([x] + z) <= (x + z)
		
                ...(but how do I manipulate ([x] + z) <= (x + z) into [x + z] = [x] + z?)
                or do I simply plug in a few values and then if all works, just say it's proven?
	

Then for all x in R and z in Z, [x + z] = [x] + z

Why are you writing your proof in a [ code] block?

 

[Melodia]

Oh because I wanted the indentation to make it look neater.

 

[sylas]

...(but how do I manipulate ([x] + z) <= (x + z) into [x + z] = [x] + z?)
or do I simply plug in a few values and then if all works, just say it's proven?
...

You don't manipulate it: you show that it satisfies the two properties. That is if y is an integer expression for which you can show

$$y \leq x \wedge (\forall z \in \mathbb{Z} : z \leq x \Rightarrow z \leq y)$$​

then from your definition statement, you can prove

$$y = \lfloor x \rfloor$$​

In this case your "x" is actually the expression x+z, and you are wanting to show [x+z] = [x]+z. So you have to simply show [x]+z satisfies the two properties.

You've obtained the first one: [x]+z <= x+z

Now you have to show

$$\forall z \in \mathbb{Z} : z \leq x \Rightarrow z \leq y$$​

except that you need the "x" to be "x+z", and you need the y to be "[x]+z". Be careful. The quantified variable used here is "z", and so you will have to change the name of that variable to avoid interfering with the z you already have in use. I suggest you change the quantified variable to "w", and then substitute the expressions for x and y.

Is this making sense?

Cheers -- sylas

 

[Mark44]

Oh because I wanted the indentation to make it look neater.

Indentation is normally used in programming to show the extent of some control structure, such as a for loop or an if ... else block. It's not primarily used in coding to make it look neater.

It makes no difference to me, though. I was just curious why you were formatting your stuff that way. Whatever floats your boat, I guess.