Consider the following construction that is used to form an instance of a set cover, taken from Vijay Vazirani (2003) Approximation Algorithms, $\S13.1.1$
Let $n=2^{k} - 1$, where $k$ is a positive integer. For $1 \le i \le n$, we use $\mathbf{i}$ to denote the $k$-bit number $i$ as a $k$-dimensional vector over GF(2).
Also define set $S_i = \{ e_j \vert \mathbf{i} \cdot \mathbf{j} = 1\}$, where $\mathbf{i} \cdot \mathbf{j}$ denotes the inner product of these two vectors. Finally, let $\mathcal{S} = \{S_1, \ldots ,S_n\}$.
The author claims that it is easy to check that each set $S_i$ contains $2^{k − 1} = (n + 1) / 2$ elements, and each element is contained in $(n + 1) / 2$ sets.
And indeed it is easy to check these claims, yet a formal proof does not appear anywhere. Am I missing something trivial here or is there a nuanced proof of this claim?