Write logic statements

Question

1) there doesn’t exist the greatest number (¬),

2) every even number is a sum of two squares, (=, ·, +),

3) there exists a number with three divisors only (=, ·),

Answers:

1) ($\forall x \in \mathbb{N} \exists y \in \mathbb{N}$) ¬($x \leq y$),

2) ($\forall x,y,p,k \in \mathbb{N}$)((x =2k)$\implies$(x =$ p·p + y·y)),

3) ($\exists x,y,p,k \in \mathbb{N}$)($(x|y)\land(p|y)\land(k|y)\land$~(x=p=k))

Are they right?

Answer 1

$$1.\ \displaystyle\mathop{\forall}_{x\in\mathbb{N}}\ \displaystyle\mathop{\exists}_{y\in\mathbb{N}}:x<y\iff\lnot(\displaystyle\mathop{\exists}_{x\in\mathbb{N}}\displaystyle\mathop{\forall}_{y\in\mathbb{N}}:x\ge y)$$ $$2.\ \displaystyle\mathop{\forall}_{n\in\mathbb{N}}\ \displaystyle\mathop{\exists}_{k,l\in\mathbb{N}}:(n=2m\implies n=k^{2}+l^{2})$$ $$3.\ \displaystyle\mathop{\exists}_{n,x_{1},x_{2},x_{3},a,b,c,d\in\mathbb{N}}:\left\{(a\not=b\not=c)\land\begin{pmatrix} a\cdot x_{1}=n\\ b\cdot x_{2}=n\\ c\cdot x_{3}=n\\ \end{pmatrix}\land [\displaystyle\mathop{\forall}_{p\in\mathbb{N}}pd=n\implies(p=a)\lor(p=b)\lor(p=c)]\right\} $$

Answer 2

Short answer: no.

1) Be careful of double negation. Try writing out "there is a greatest number," and then negating that. Remember the standard technique of swapping $\forall$ and $\exists$, though you cannot do this wantonly.

Also, a notation tip: when you are talking about $\mathbb{N}$, first go to $n, m$. You did good by using $p,k$, but using $x,y$ was confusing in $\mathbb{N}$. See Terry Tao's blog post "Use good notation".

2) You are essentially saying that every number is simultaneously the square of every integer. You need to have an existence statement.

3) You don't appeal to the part of the question that says "three divisors only". You say that there are three divisors but you don't say that there cannot be more. I'm also assuming that you mean ~ to be $\lnot$, in which case, it could be true that $x = y$ but if $x \neq k$, then $k \neq y$, so $x \neq k \neq y$ is true, but there are not three unique divisors.

Also, feel free to use subscripts to make your notation cleaner. So if you want to say that $n$ has only three divisors, you could call them $d_1, d_2, d_3$, rather than $y$ having the three divisors $x, p,k$.