Finding specific elements in a finite set

Question

Given the set $A = \{0, 1\}^8$, how can I find the set of all elements in A with exactly 4 zero entries?

Answer 1

A suitable notation would be

$$\{(x_{1},...,x_{8})\in A\ | \ \sum_{i=1}^{8}{x_{i}}=4\}$$

Answer 2

There are many ways to describe the set in question. One that hews very closely to the language in which you described it is

$$\left\{\varphi\in A:\left|\varphi^{-1}\big[\{1\}\big]\right|=4\right\}\;.$$

Here I use the fact that elements of $\{0,1\}^8$ are functions from some $8$-element set to $\{0,1\}$. If you prefer to think of them as $8$-tuples, you may prefer other descriptions, e.g.,

$$\left\{\langle a_1,\dots,a_8\rangle\in\{0,1\}^8:\left|\big\{k\in\{1,\dots,8\}:a_k=1\big\}\right|=4\right\}$$

or, with a shorter but less direct translation,

$$\left\{\langle a_1,\dots,a_8\rangle\in\{0,1\}^8:\sum_{k=1}^8a_k=4\right\}\;.$$