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For each statement given, I had to write a negation then decide which of the two is true, would anyone check them for me?

Statement: For each $n∈\mathbb{N}$, there is $m∈\mathbb{N}$ such that $n\leq m$. TRUE

Negation: There exists $n∈\mathbb{N}$ such that for each $m∈\mathbb{N}$, $n>m$.

Statement: There is $n∈\mathbb{N}$ such that for all $m∈\mathbb{N}, n\leq m$. TRUE

Negation: For each $n∈\mathbb{N}$, there exists $m∈\mathbb{N}$ such that $n> m$.

Statement: There is $m∈\mathbb{N}$ such that for all $n∈\mathbb{N}, n\leq m$. TRUE

Negation: For each $m∈\mathbb{N}$, there exists $n∈\mathbb{N}$ such that $n>m$.

Statement: For each non-zero $x∈\mathbb{R}$, there is $y∈\mathbb{R}$ such that $xy=1$. TRUE

Negation: There exists non-zero $x∈\mathbb{R}$, such that for all $y∈\mathbb{R}$ $xy\ne1$.

Statement: There is $x∈\mathbb{R}$ such that for all $y∈\mathbb{R}$, $xy=1$.

Negation: For each $x∈\mathbb{R}$, there exists $y∈\mathbb{R}$, $xy\ne1$.

Statement: There is $x∈\mathbb{R}$ such that for all $y∈\mathbb{R}$, $xy=0$. TRUE

Negation: For each $x∈\mathbb{R}$, there exists $y∈\mathbb{R}$ such that $xy\ne0$.

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    The negation of "for each $n\in\mathbb{N}$, blah blah" is "there exists $n\in\mathbb{N}$ such that NOT[blah blah]" ... – 2017-02-23
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    Ah thanks. So for example the statement: For each $n∈\mathbb{N}$, there is $m∈\mathbb{N}$ such that $n\leq m$, the negation should be: There exists $n∈\mathbb{N}$ such that there isn't $m∈\mathbb{N}, n\leq m$? – 2017-02-23
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    Close vote: why should anyone on MSE filter your homework answers before you get them marked? – 2017-02-23
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    @MikeA: Also, the negation of "there exists $m\in\mathbb{N}$ such that ..." is "for each $m\in\mathbb{N}$, NOT[...]". – 2017-02-23
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    No-one should feel obligated and I'm sure no-one does feel obligated to help me out. If someone does take the time to do so, then that is kind of them and I appreciate it. – 2017-02-23
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    @Adren Nearly all of them need a reevaluation by myself, I wasn't nearly stringent enough with the logic, cheers for pointing these out. – 2017-02-23
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    @MikeA: You are welcome. Now try to adjust everything and edit the OP. – 2017-02-23
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    @Adren will do, I was wondering whether I should edit the post. – 2017-02-23
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    @MikeA: anyone who feels inclined to filter your homework answers is providing a homework help service that isn't what I thought MSE was for. – 2017-02-23
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    @Rob Arthan Well I've learnt something about logic and negating statements, and ultimately the site is about educating people, unless my interpretation is wrong. – 2017-02-23
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    @MikeA: so what would be the educational value to you of a literal answer to your question? MSE is **not** a homework answering service. – 2017-02-25
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    @RobArthan I have attempted a question for which I don't have access to the correct answer. If my answer is correct and I don't know that is the case, then I can't have confidence that I have understood the material covered in said question. If my answer is incorrect, then knowing it is a mistake allows me the opportunity to try and learn from it, which I can't do if I don't know it is wrong. That is the educational value of a literal answer and that is not including other advice I have gained in the thread that wasn't a literal answer. – 2017-02-25

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For simple statements like these, you can develop an intuition by forming them in natural language, negating that, and then rephrasing the negation mathematically. To use your first problem,

For each $n \in N$, there exists $m \in N$ such that $n < m$.

This is saying

There's always a larger natural number.

The negation of which is

There is a largest natural number

Which is expressed mathematically as

There exists $n \in N$ such that for all $m \in N$, $n \ge m$.

Try this with your other statements. You should notice a pattern in how a statement relates to its negation. Additionally, phrasing it in natural language will probably help you figure out its truth value.

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    Thanks for you comment, that's a very helpful explanation! I'm gonna work through these again with this in mind. Is this a good start? Statement: There is $n∈\mathbb{N}$ such that for all $m∈\mathbb{N}, n\leq m$. Negation: For each $n∈\mathbb{N}$, there exists $m∈\mathbb{N}$ such that $n>m$. So the statement in plain english would be: There is a smallest natural number and plain english negation: There's always a smaller number. – 2017-02-23