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I couldn't be sure about the negation of the statement $\exists x \in N , x < 3$

is it $$ \lnot(\exists x \in N , x < 3) \equiv (\forall x \in N , x \geq 3) $$ or $$ \lnot (\exists x \in N , x < 3) \equiv (\forall x \notin N , x \geq 3) $$ can someone help me (with an explanation)?

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This might be helpful. –  Asaf Karagila Oct 1 '12 at 13:54

3 Answers 3

$$ \begin{align*} \lnot (\exists x \in N, x < 3) &\equiv \forall x \in N, \lnot(x < 3)\\ &\equiv \forall x \in N, x \geq 3 \end{align*} $$

Think about what the sentence is saying in words. The original claim is

$N$ contains a number smaller than 3.

Imagine someone says this to you, but you're certain that person is mistaken. What would you have to show them in order to negate the claim? In words, you might say "You are mistaken. The truth is:

$N$ does not contain a number smaller than 3.

In other words:

Every number belonging to $N$ is greater than or equal to 3.

You would most certainly not want to start talking about numbers not belonging to $N$. The person you wish to refute made no claim about such numbers, so neither should your refutation involve them.

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just to be sure, to me your statement reads as "There exists an $x$, element of the positive integers (doesn't matter with or withour $0$), such that $x$ is less than $3$. If this is correct, then the negation is: There is no $x$ element of the positive integers such that $x$ is less than $3$. This means that all $x$ element of the positive integers are greater equal $3$.

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When we write $\forall x\in A:\varphi(x)$ we actually abbreviate $\forall x(x\in A\rightarrow \varphi(x))$; and when we write $\exists x\in A:\varphi(x)$ we abbreviate $\exists x(x\in A\land\varphi(x))$.

Two useful equivalences I will use are

  1. $\lnot(p\land q)\iff(\lnot p\lor\lnot q)$, and
  2. $p\rightarrow q\iff\lnot p\lor q$.

Now recall that $\lnot\exists x\varphi(x)$ is equivalent to $\forall x\lnot\varphi(x)$. Use both and we have:

$$\begin{align} \lnot(\exists x\in N:x< 3) &\iff\lnot(\exists x(x\in N\land x< 3))\\ &\iff\forall x(\lnot (x\in N\land x< 3))\\ &\iff\forall x(x\notin N\lor x\geq 3)\\ &\iff\forall x(x\in N\rightarrow x\geq 3)\\ &\iff\forall x\in N:x\geq 3 \end{align}$$

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I mostly got the idea and the explanation but I have one other question, at the beginning why did you expressed ¬(∃x∈N:x<3)⟺¬(∃x(x∈N∧x<3)) using ∧ and then at the end you did ∀x(x∈N→x≥3) ⟺ (∀x∈N:x≥3). By doing so you transformed → into semicolon. Is semicolon ∧ or → ?? –  Deniz Önal Oct 7 '12 at 1:54
@Deniz: Neither. The $:$ is what you used a comma for. I just really hate commas in formal sentences. –  Asaf Karagila Oct 7 '12 at 6:44

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