How do I properly turn a statement into a mathematical proposition?

Given the statement: Every natural number is odd

I have to turn this statement into a mathematical proposition. I also have to write up its negation as well (where the $\neg$ doesn't show up in the final answer).

Steps I took:

$$N(x) : x \in\mathbb{N},\quad k \in\mathbb{Z}$$

$$\forall x(N(x)\rightarrow \exists k(N(k)\wedge x=2k+1))$$

Negation: $\neg \forall x(N(x)\rightarrow \exists k(N(k)\wedge x=2k+1))$

$$\Rightarrow\exists x\neg (N(x)\rightarrow \exists k(N(k)\wedge x=2k+1))$$

$$\Rightarrow\exists x (N(x)\wedge\neg \exists k(N(k)\wedge x=2k+1))$$

$$\Rightarrow\exists x (N(x)\wedge \forall k \neg (N(k)\wedge x = 2k+1))$$

$$\Rightarrow\exists x (N(x)\wedge \forall k(N(k)\rightarrow x\neq 2k+1))$$

I'm almost positive about my answer, but my problem is that I don't truly understand what I did here because I just went off or things I saw in the MIT OCW lecture notes to try and formulate this. One of the drawbacks to self studying is not being able to get something clarified when I don't understand. I am hoping that someone here can break down a few things for me here (in layman terms).

Why was $N(x)$ used? Why couldn't I just use $x\in\mathbb{N}$ ? I don't understand why I had to create a function in order to represent the $\mathbb{N}$. I would also like understand the predicate to this statement. I wrote it up that way because another example contained a similar format. I would like to know why it had to use the $\wedge$ operator.

Please break it down to me in layman terms. I'm not used to this type of math and MIT's Math for CS is my first exposure to Discrete Math. (Although I know that some people here might argue that it's barely scratching the surface of discrete math).

• Small typo, the second right parenthesis should be erased. Nov 1 '15 at 18:38
• @AndréNicolas Would you please be able to point out where exactly? Nov 1 '15 at 18:41
• Second displayed formula starts with $\forall x(N(x))\to$. It should start with $\forall x(N(x)\to$. Nov 1 '15 at 18:45
• The last step is wrong; from $¬∃k(N(k)∧x=2k+1)$ to $∀k¬(N(k)∧x=2k+1)$ to $∀k(N(k)→x≠2k+1)$. Nov 1 '15 at 18:56
• @AndréNicolas Why put a parentheses at all then? Would you be able to explain why $N(x)$ is used represent the natural numbers? Nov 1 '15 at 19:03

It strikes me that you could just write $\forall x\in \mathbf{N} \exists k\in \mathbf{N}\cup \{0\} \hspace{0.5cm} \mbox{such that} \hspace {0.5cm} x=2k+1$
• The problem specifically stated that I need to start by putting a $\neg$ outside of the initial statement and work it inside. Nov 1 '15 at 18:41