How to write such list in set builder notation?

Question

I'm pretty new with mathematical proofing and came across this question. How do you write $$B = \left\{16,8,4,2,1, \dfrac{1}{2}, \dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{16}\right\}$$ in set builder notation?

Answer 1

$$B=\{2^n:n\in\Bbb Z,|n|<5\}$$ $$=\{2^n:n\in\Bbb Z,|n|\le 4\}$$ $$=\left\{\left(\dfrac{1}{2}\right)^n:n\in\Bbb Z,|n|<5\right\}$$ $$=\left\{\left(\dfrac{1}{2}\right)^n:n\in\Bbb Z,|n|\le4\right\}$$ where $\Bbb Z$ denotes the set of all integers. There are infinitely many ways to represent it in set-builder notation. Another interesting one $$B=\{x:\log_2x\in\Bbb Z,\vert\log_2x\vert<5\}$$ $$=\{x:\log_2x\in\Bbb Z,\vert\log_2x\vert\le 4\}$$

Answer 2

There is no unique answer.

The most direct way to do so is: $$B=\left\{x\middle|x=\tfrac{1}{16}\lor x=\tfrac{1}{8}\lor x=\tfrac{1}{4} \lor x=\tfrac{1}{2}\lor x=1 \lor x=2 \lor x=4 \lor x=8 \lor x=16\right\}$$ This is literally what the list notation means, and therefore it works no matter what elements are in your list.

Here $\lor$ is the symbol for logical or, that is, the above says:

$B$ is the set of all the $x$ such that $x=\frac{1}{16}$ or $x=\frac{1}{8}$ or … or $=8$ or $x=16$

Now obviously for the special case there are relations between the elements which you can use to get a simpler expression. In particular, it is easily seen that all of the numbers are powers of two, therefore a simpler way to write it is $$B=\left\{2^n\middle|n\in\mathbb Z\land -4\le n \le 4\right\}$$

Here $\land$ is the symbol of logical and. Often a comma is written instead, so you'd get $$B=\left\{2^n\middle|n\in\mathbb Z, -4\le n \le 4\right\}$$

Also note that there are different conventions as to what to put between the left and right side of the set builder expression. The vertical bar I've used here is a quite common choice, another common choice is the colon: $$B=\left\{2^n: n\in\mathbb Z\land -4\le n \le 4\right\}$$

Also note that there are really two forms of set builder notation that are used side by side. The first one has on the left of the bar (or colon) the expression, and on the right side the condition its variables has to fulfil. All of the above notations are of that form. But there's also a second form, where on the left you learn which larger set those elements are from, and on the right you learn which restrictions you have to apply to cut down that larger set to your set.

In that form, $B$ might be written as follows: $$B=\left\{x\in\mathbb Q\setminus\{0\}\middle|16x\in\mathbb N\land \frac{16}{x}\in\mathbb N\right\}$$