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)

Re: prove or disprove question about two natural number

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**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 $\displaystyle \{1,2\}\subset\mathbb{N}~\&~1<2$, what natural number is between them?

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

Re: prove or disprove question about two natural number

Quote:

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.

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 .

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Re: prove or disprove question about two natural number

I hope maybe this will help in general.

Attachment 27346

Re: prove or disprove question about two natural number

Quote:

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 $\displaystyle f(x)=|x|$ is continuous but has no derivative at $\displaystyle x=0$.

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.