What is the actual way of negating a mathematical statement?

Question

For each $x \gt 0$ and $y \in \Bbb R,$ there exists $n \in \Bbb N$ such that $nx \gt y.$

How to negate the above statement? I am doing in the following way $:$

There exists $x \leq 0$ or $y \notin \Bbb R$ such that for all $n \in \Bbb N$ we have $nx \leq y.$

But I don't think it's a valid negation. Actually I don't know what's the actual way to negate a statement. Is there any certain rule which can be used in order to negate any mathematical statement? Also I have just heard the word "quantifiers". Can anybody please clear it to me about the role of "quantifiers" here? What does that actually mean?

Any suggestion regarding this will be appreciated. Thanks.

Answer 1

Your proposition should be written as

$$(\forall (x,y)\in \Bbb R^+\times \Bbb R)\; (\exists n\in \Bbb N) \;:$$ $$\;nx>y$$

and its negation will be

$$(\exists (x,y)\in \Bbb R^+\times \Bbb R)\;:\;(\forall n\in \Bbb N)\;$$ $$nx\le y$$

Answer 2

A condition $P(x)$ on a quantified variable $x$ can be moved to the statement following the quantifier as follows: $$\begin{align} \forall x\ P(x) : Q(x) &\quad\equiv\quad \forall x : P(x) \implies Q(x) \\ \exists x\ P(x) : Q(x) &\quad\equiv\quad \exists x : P(x) \land Q(x) \end{align}$$ For example, $$\begin{align} \forall x>0 : Q(x) &\quad\equiv\quad \forall x : x>0 \implies Q(x) \\ \exists x\in \mathbb{N} : Q(x) &\quad\equiv\quad \exists x : x\in\mathbb{N} \land Q(x) \end{align}$$

By the laws of negating quantifiers, $$\begin{align} \neg\forall x \text{ s.t. }P(x) : Q(x) &\quad\equiv\quad \exists x : \neg(P(x) \implies Q(x)) \\ \neg\exists x \text{ s.t. }P(x) : Q(x) &\quad\equiv\quad \forall x : \neg(P(x) \land Q(x)) \end{align}$$

Now remember the negation laws of implication and conjunction: $$\begin{align} \neg(P \implies Q) &\quad\equiv\quad P \land \neg Q\\ \neg(P \land Q) &\quad\equiv\quad \neg P \lor \neg Q \end{align}$$

Thus, $$\begin{align} \neg\forall x \ P(x) : Q(x) &\quad\equiv\quad \exists x : P(x) \land \neg Q(x) &\quad\equiv\quad \exists x\ P(x) : \neg Q(x) \\ \neg\exists x \ P(x) : Q(x) &\quad\equiv\quad \forall x : \neg P(x) \lor \neg Q(x) &\quad\equiv\quad \forall x\ P(x) : \neg Q(x) \end{align}$$

The given statement "For each $x \gt 0$ and $y \in \Bbb R,$ there exists $n \in \Bbb N$ such that $nx \gt y$" can be written $$ \forall x>0\ \forall y\in\mathbb{R}\ \exists n\in\mathbb{N} : nx>y $$ so the negation is $$ \exists x>0\ \exists y\in\mathbb{R}\ \forall n\in\mathbb{N} : nx\leq y $$

Answer 3

Technically, the negation of this statement is just

It is not the case that for each $x \gt 0$ and $y \in \Bbb R,$ there exists $n \in \Bbb N$ such that $nx \gt y.$

-- but usually these kinds of exercises expect you to push the negation inside as far as possible to arrive at a logically equivalent ($\equiv$) statement.

Quantifiers are operators that, well, quantify elements. "for all" and "there exists" are quantifiers. ("for all" and "for each" mean the same.) Other examples of quantifiers include "no", "not all", "exactly 2" or "at least 3".
Quantifiers often come with a so-called restriction, which limits which objects the quantifier ranges over, e.g. the $x > 0$ after the "for each" and the "$y \in \mathbb{R}$" after the "and" in your example.
In general, you don't negate the restriction of the quantifier, but only the part that follows it. So you don't e.g. turn $x > 0$ into $x \leq 0$, because that's just part of the restriction, but instead you pass the negation to the part after "for each $x > 0$".

The rules for negating quantifiers are as follows:

• not (for all x) $\equiv$ there exists an x such that not
• not (there exists an x) $\equiv$ for all x not
• not (no x) $\equiv$ there exists an x such that

The rules for pushing negations of logical operators inside are:

• not (A and B) $\equiv$ (not A) or (not B)
• not (A or B) $\equiv$ (not A) and (not B)
• not (if A then B) $\equiv$ (A) and (not B)
• not (A if B) $\equiv$ (not A) and (B)
• not (A if and only if B) $\equiv$ ((A) and (not B)) or ((not A) and (B))

In addition, there are special negations of some arithmetic operations:

• not ($x = y$) $\equiv$ $x \neq y$
• not ($x \geq y$) $\equiv$ $x < y$
• not ($x > y$) $\equiv$ $x \leq y$

Double negations get eliminated:

• not not A $\equiv$ A

Finally, quantification with "and" is in fact two nested quantififactions:

• for all x and y $\equiv$ for all x for all y
• there exists x and y such that $\equiv$ there exists x such that there exists y such that

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And now just apply these rules to your sentence step by step -- this is a purely mechanical procedure, just follow the negation through until there are no more rules to apply.

not for each $x \gt 0$ and $y \in \Bbb R,$ there exists $n \in \Bbb N$ such that $nx \gt y$
$\equiv$ not for each $x \gt 0$ for each $y \in \Bbb R$ there exists $n \in \Bbb N$ such that $nx \gt y$
$\equiv$ there exists $x \gt 0$ such that not for each $y \in \Bbb R$ there exists $n \in \Bbb N$ such that $nx \gt y$
$\equiv$ there exists $x \gt 0$ such that there exists $y \in \Bbb R$ such that not there exists $n \in \Bbb N$ such that $nx \gt y$
$\equiv$ there exists $x \gt 0$ such that there exists $y \in \Bbb R$ such that for all $n \in \Bbb N$ not $nx \gt y$
$\equiv$ there exists $x \gt 0$ such that there exists $y \in \Bbb R$ such that for all $n \in \Bbb N$ $nx \pmb{\pmb{\leq}} y$
$\equiv$ there exists $x \gt 0$ and $y \in \Bbb R$ such that for all $n \in \Bbb N$, $nx \leq y$

In semi-formal notation:

$\exists x \gt 0\ \exists y \in \Bbb R\ \forall n \in \Bbb N: nx \leq y$

In formal notation:

$\exists x (x \gt 0 \land \exists y (y \in \Bbb R \land \forall n (n \in \Bbb N \implies nx \leq y)))$