I am required to give a (propositional) formula $\varphi_n$ for every n with $vars(\varphi_n)={X_0,...,X_{2n-1}}$ so that the following holds:
$$\sum_{i<n}\beta(X_i)2^i \leq \sum_{i<n}\beta(X_{n+i})2^i \iff \beta:\{X_0,...,X_{2n-1}\} \rightarrow \{0,1\} \text{ is a satisfying assignment for } \varphi_n$$
I looked at the solution for that problem. The solution says: $$\varphi_n=\underbrace{\bigwedge\limits_{j=0}^{n-1}(X_j\leftrightarrow X_{j+n})}_{=} \text{ }\lor\text{ }\underbrace{\bigvee\limits_{i=0}^{n-1}\left( (\lnot{X_i}\text{ }\land\text{ } X_{i+n})\land \bigwedge\limits_{j=i+1}^{n-1}(X_j \leftrightarrow X_{j+n})\right)}_{<}$$
That looks way better than my approach:
$$n=1: \varphi_1 = \lnot( X_0 \land \lnot X_1)\\ n\rightarrow n+1: \varphi_{n+1} = (\lnot\varphi_n \land X_{2n+1} \text{ }\land \lnot X_n)\lor(\varphi_n \text{ } \land \text{ } \lnot(X_n \text{ }\land \text{ }\lnot X_{2n+1}) )$$
But would by mine correct, too?