If a set is represented by curly bracket $\{\ \ \}$
Then why is a closed set represented by $[\ \ ]$ and open set represented by $(\ \ )$
If a closed set is represented by $[\ \ ]$ , then is a vector
$$x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$$
also a set? E.g., a set containing $x_1, x_2$.
On
It's just notation, you shouldn't think so hard about it. At some point we were gonna run out of symbols.
In any case, if you have set theory as a foundation, everything is a set formally by definition, but this might not be the healthiest view to have if one is not careful.
So yes, in this case a vector is a set, but don't worry about it. Let's say vectors are not sets instead. In your particular case where you have a vector for a particular base, the key difference from a set is that the elements are ordered and can be repeated.
e.g: $\qquad$ $\{a,a,b\} = \{b,a\}$ but $[a,a,b] \neq [b,a]$
The other symbols $[\ ]$ and $(\ )$ are used because they represent something visually. When you use $[a,b]$ you mean the closed interval from $a$ to $b$, but $(a,b)$ is the open interval from $a$ to $b$.
The shapes of the different brackets suggest this behavior. We used what we had, it's not a perfect representation but it's ok for the symbols we had available.
But again, symbols in math get reused all the time. When you see $[x]$ to mean the equivalence class of $x$, this has nothing to do with closed intervals. When you see $\pi$ this might be used as a projection, not necessarily the number pi, and so on and on.
Hope this helps. Cheers!
On
These are notation using limited SYMBOLS with reuse.
Braces are used for Sets like $A=\{1,2,3,4,5,6,7,8\}$ where we can write out al the elements.
Square Brackets are used for Closed Sets like $B=[0,1]$ which is a short cut to write out the elements with Braces like $B=\{0.0,0.1,\cdots,0.22,0.033,\cdots,0.4,0.567,0.8,\cdots,0.99999,\cdots,1.0\}$. Can we write out all the elements ? No way ! Hence , we use Square Brackets to mean all numbers between the 2 limits.
If we had used Braces like $C=\{0,1\}$ , it is not the list of all numbers between $0$ & $1$ , it is just 2 numbers , nothing more.
Like-wise , Open Set $D=\{0.000001,0.1,\cdots,0.22,0.033,\cdots,0.4,0.567,\cdots,0.8,\cdots,0.999, 0.99999\}$ , where we are leaving out the 2 limits. We can not write out the elements.
It is a shortcut to write it $D=(0,1)$
Coming to vector notation , whether we use Square Brackets or regular Brackets , whether we write it column-wise or row-wise , it is not a Set because the Order matters for the vector.
These are all notations where we can easily Differentiate between the usages.
We have limited SYMBOLS hence we have to reuse them.
Compare :
We are given $d=a+bc$ & $dy/dx=e^x$ : Can we cancel the $d$ values when we Differentiate ? No , it is a notation & $d$ is not the earlier variable.
BTW , why the word Differentiate has at least two meanings here ? We have limited words ( & SYMBOLS ) hence we have to reuse them.
The notation you are referring to is pretty much specific to intervals in $\Bbb R$ with the usual topology. This is a pretty specific setting. Open and closed sets are concepts in topology, and for a general topological space, there is no notion of closed or open intervals, or even intervals at all.
In $\Bbb R$ with the usual topology, $[a,b]$ is just shorthand for $\{x\in\Bbb R:a\le x\le b\}$. Similarly, we write $$(a,b)=\{x\in\Bbb R:a<x<b\},$$ but the exact same notation $(a,b)$ in another context means a single point in $\Bbb R^2$.
In the example you gave, the square brackets in $$\begin{bmatrix}{x_1\\x_2}\end{bmatrix}$$ only indicate that the object is a vector. Formally, this vector would probably be considered a pair $(x_1,x_2)\in\Bbb R^2$, which is not to be confused with the set containing that pair, i.e., $\{(x_1,x_2)\}$. If you want to talk about open and closed subsets of $\Bbb R^2$, you have to know precisely what that means first, since you can no longer even refer to intervals.
Plenty of authors use rounded parenthesis for vectors and would write the exact same vector as $$\begin{pmatrix}{x_1\\x_2}\end{pmatrix}.$$ In combinatorics, we typically interpret the above notation instead as a binomial coefficient: $$\begin{pmatrix}{m\\n}\end{pmatrix}=\frac{m!}{n!(m-n)!}.$$ My point is that it's okay for notation to be reused as long as the precise meaning is clear from context.
Returning to my first sentence, I want to emphasize that "open" and "closed" are concepts in topology, not set theory. Given an arbitrary set, there are no open or closed sets at all until you define a topology. Even when you have a topological space, the open and closed sets can be quite weird, and you can't always rely on open and closed intervals of $\Bbb R$ for intuition.