How do I negate the following statement? How do I negate the following statement?
What are the steps which should be taken to negate an argument such as the one given?
$$\exists y \forall x ((y > 0) \land (x < y))$$
 A: Essentially the $\forall$ becomes $\exists$ and vice-versa, and $\land$ becomes $\lor$. Hence, we have:
$$\begin{align}\lnot\left(\exists y\forall x((y>0)\land(x<y))\right)&\iff\forall y\exists x(\lnot(y>0)\lor\lnot(x<y))\\&\iff\forall y\exists x((y\leq 0)\lor(x\geq y))\end{align}$$
A: Remember, when you negate a predicate, you negate "each part of it". It's like the negation distributes.
Some rules are: $$\neg\exists=\forall, \neg\forall=\exists,\neg(\lor)=\land, \neg(\land)=\lor,\neg P=\neg P$$
Therefore, distribute the $\neg$
$$\neg(\exists y \forall x ((y > 0) \land (x < y)))\iff \forall y\exists x(\neg((y > 0) \land (x < y))$$
Now, $\neg(y\gt 0)=(y\leq 0)$, "if its not greater, then it must be less than or equal to"
Similarly, $\neg(x<y)=(x\geq y)$
Finishing out negation off, we get:
$$\iff\forall y \exists x ((y\leq 0)\lor(x\geq y))\text{ notice $\land$ changes to $\lor$}$$
A: Theorem. The negation of $$\exists y \forall x ((y > 0) \land (x < y))$$ is $$\forall y \exists x ((y \le 0) \lor (x \ge y))$$ 
Proof.
By Lemma 1, 2, 3, 4, and 5.

Lemma 1. The negation of $\exists y~ P(y)$ is $\forall y,~\lnot P(y)$

Proof.
Calling $Y$ the universe
$$\exists y~ P(y)\equiv\exists y\in Y,~ P(y)\equiv Y\cap P\neq\emptyset$$
where $P=\{z|~P(z)\}$.
So its negation is: $$Y\cap P=\emptyset\equiv Y\subset P^c\equiv\forall y\in Y,~\lnot P(y)$$
where $P^c=\{z|~\lnot P(z)\}$

Lemma 2. The negation of $\forall y~ P(y)$ is $\exists y,~\lnot P(y)$

Proof.
Calling $Y$ the universe
$$\forall y~ P(y)\equiv\forall y\in Y,~ P(y)\equiv Y\subset P$$
So its negation is: $$Y\not\subset P\equiv Y\cap P^c\neq\emptyset\equiv\exists y\in Y,~\lnot P(y)$$

Lemma 3. The negation of $a\land b$ is $\bar a\lor\bar b$

Proof.
By de Morgan $$a\land b=\overline{\bar a\lor\bar b}$$

Lemma 4. The negation of $y>0$ is $y\le 0$

Proof.
Here it is understood that $y\in \mathbb{R}$, though not explicitly stated.
$$y>0\equiv y\in\{z|~z>0\}$$
So its negation is
$$y\not\in\{z|~z>0\}\equiv y\in\{z|~z>0\}^c$$
where $\{z|~z>0\}^c=\mathbb{R}\setminus\{z|~z>0\}=\{z|~z\le0\}$

Lemma 5. The negation of $x<y$ is $x\ge y$

Proof.
Here it is understood that $x,~ y\in \mathbb{R}$, though not explicitly stated.
$$x<y\equiv (x,y)\in\{(w,z)|~w< z\}$$
So its negation is
$$(x,y)\not\in\{(w,z)|~w< z\}\equiv y\in\{(w,z)|~w< z\}^c$$
where $\{(w,z)|~w< z\}^c=\mathbb{R}^2\setminus \{(w,z)|~w< z\}=\{(w,z)|~w\ge z\}$
