The set of Fibonacci numbers formalized in set-theoretic notation: did I do it correctly? The set of Fibonacci numbers $= \displaystyle \{x_i \in \Bbb N_0 : x_i = x_{i-1} + x_{i-2} \ \forall \ i \in \Bbb N_2 : x_0 + x_1 =1, >\}$.
Is this correct notation? The $>$ at the end is supposed to show it is an ordered set, ordered by the "greater than" operator.
EDIT:
The way I wrote it may have been a little confusing. Here's an alternative:
$$\{x_i \in \Bbb N_0 : x_i = x_{i-1} + x_{i-2} \ \forall \ i \in \Bbb N_2 : x_0 = 0 \land x_1 =1, >\}$$
However, since we're dealing with naturals in an ordered set, and since the values are $0$ and $1$, the values of $x_0$ and $x_1$ can be conveyed with their sum.
 A: Let us see what are the problem with your notation. First you wrote:
$$\displaystyle \{x_i \in \Bbb N_0 : x_i = x_{i-1} + x_{i-2} \ \forall \ i \in \Bbb N_2 : x_0 + x_1 =1, >\}$$
But notice $x_0+x_1=1$ is not what you want to say, since $x_0=1,x_1=0$ would work in this definition but this is not what you want. Then you wrote:
$$\{x_i \in \Bbb N_0 : x_i = x_{i-1} + x_{i-2} \ \forall \ i \in \Bbb N_2 : x_0 = 0 \land x_1 =1, >\}$$
So you solved this problem, but there are some more thing to correct. $\mathbb{N}_0$ is a notation for $\mathbb{N}\cup\lbrace 0\rbrace$, that is ok. However $\mathbb{N}_2$ is not a good notation, what you want to say is ''all natural numbers greater or equal than 2'', the standard notation for this is $i\in\mathbb{N}$ such that $i\geq 2$. Sometimes, when there is no doubt that we are working in the natural numbers (such as in this case) you can ommit $i\in\mathbb{N}$ and write only $i\geq 2$. Solving this problem, you will get:
$$\{x_i \in \Bbb N_0 : x_i = x_{i-1} + x_{i-2} \ \forall \ i \geq2  : x_0 = 0 \land x_1 =1, >\}$$
Another problem is that you cant use two times $:$ in the definition of a set. In set theory a set is written as ''{someting : conditions on that something}'', so the correct way to write this would be:
$$\{x_i \in \Bbb N_0 : x_i = x_{i-1} + x_{i-2} \ \forall \ i \geq2 , x_0 = 0 \land x_1 =1, >\}$$
That would be better, however notice that you are defining $x_2=x_1+x_0$ before saying what $x_0$ and $x_1$ is, so it is better to change the order. Also no need to use $\land$, since it there is no ambiguity (if you say $x_0=0,x_1=1$ everyone will understand you). Correcting this you will have:
$$\{x_i \in \Bbb N_0 : x_0 = 0, x_1 =1, x_i = x_{i-1} + x_{i-2} \ \forall \ i \geq2 , >\}$$
The last thing I want to change is the $x_i \in \Bbb N_0$, since you know that the Fibonnaci numbers are a countable subset of the real numbers, you have to specify this before the two dots. I will denote it as $\lbrace x_n\rbrace_{n=0}^{\infty}\in\mathbb{N}_0$. Hence:
$$\{\lbrace x_n\rbrace_{n=0}^{\infty}\in\mathbb{N}_0: x_0 = 0, x_1 =1, x_n = x_{n-1} + x_{n-2} \ \forall \ n \geq2 , >\}$$
One last remark. The notation for $>$ at the end does not look good to me, maybe it is a particular notation from your teacher or you read it somewhere, but I've never seen these particular notation. Apart from this, this is a subset of the natural numbers, which is obviously ordered, so I wouldn't write it. The way I would have written it is:
$$\{\lbrace x_n\rbrace_{n=0}^{\infty}\in\mathbb{N}_0: x_0 = 0, x_1 =1, x_n = x_{n-1} + x_{n-2} \ \forall \ n \geq2 \}$$
Hope this helps. It seems that you are having problems with basic notation and set theory, but hope this makes you understands what are the problems on your notation and how to sove them.
