1. ## prove or disprove question about two natural number

prove or disprove : for all M, N in natural number , M < N => (exists some number K in natural number ,such that M < K and k < N)
I notice that this is a false statement which i need to proof the negation of statement to disprove it.
so far i got ,

for all M , N in natural number , M < N => (for all number K in natural number , that M => K or K <= N)

2. ## Re: prove or disprove question about two natural number

Originally Posted by amazingboxz
prove or disprove : for all M, N in natural number , M < N => (exists some number K in natural number ,such that M < K and k < N)
I notice that this is a false statement which i need to proof the negation of statement to disprove it.
so far i got , for all M , N in natural number , M < N => (for all number K in natural number , that M => K or K <= N)
Well $\{1,2\}\subset\mathbb{N}~\&~1<2$, what natural number is between them?

3. ## Re: prove or disprove question about two natural number

no natural number can be exists in between 2 Consecutive Numbers , that is why i think this is false statement and need to be disprove

4. ## Re: prove or disprove question about two natural number

Originally Posted by amazingboxz
no natural number can be exists in between 2 Consecutive Numbers , that is why i think this is false statement and need to be disprove
Don't you understand that disproves the statement?
One counter-examples is sufficient to disprove a statement.

5. ## Re: prove or disprove question about two natural number

here is the problem , my prof haven't mention any thing about disprove the statement using counter example and I'm not sure we are allow to use counter-example in this assignment that why i'm struggling.
Anyway thanks for the help , very appreciated

P.S , the reason is because in this course , we are required to construct a proof structure to obtain full mark. which means that we can't just simply provide a counter-example .

6. ## Re: prove or disprove question about two natural number

I hope maybe this will help in general.

7. ## Re: prove or disprove question about two natural number

Originally Posted by amazingboxz
here is the problem , my prof haven't mention any thing about disprove the statement using counter example and I'm not sure we are allow to use counter-example in this assignment that why i'm struggling.
Anyway thanks for the help , very appreciated
P.S , the reason is because in this course , we are required to construct a proof structure to obtain full mark. which means that we can't just simply provide a counter-example .
Surely none of the above is true. No competent mathematics professor would have failed to point out a basic fact of proof-theory. You must have misunderstood the professor.

Did you miss class the day basic proof-theory was explained? Surely you have heard it said: "we cannot prove a negative"?
However, we can disprove a positive by giving one counter-example. If someone says "all continuous functions have derivatives", you disprove the statement pointing out $f(x)=|x|$ is continuous but has no derivative at $x=0$.

Now if I am not incorrect about your take on this instructor, then you need to have a sit down discussion with the department chair about this. As a retired university mathematics chair I can tell how important that can be. BUT don't do it it if your facts are not in order.