Consider truth assignments involving only the propositional variables $x_0, x_1, x_2, x_3$ and $y_0, y_1, y_2, y_3$.
Every such truth assignment gives a value of $1$ (representing true) or $0$ (representing false) to each variable. We can therefore think of a truth assignment $\tau$ as determining a four-bit integer $x_\tau$ depending on the values given to $x_0, x_1, x_2$ and $x_3$, and a four-bit integer $y_\tau$ depending on the values given to $y_0, y_1, y_2$ and $y_3$.
More precisely, with $\tau (x_i)$ being the truth value assigned to $x_i$, we can define the integers $x_\tau = 2^3 \tau (x_3) + 2^2 \tau (x_2) + 2^1 \tau (x_1) + \tau (x_0)$ and $y_\tau = 2^3\tau (y_3) + 2^2 \tau (y_2) + 2^1 \tau (y_1) + \tau (y_0)$.
Write a formula that is satisfied by exactly those truth assignments $\tau$ for which $x_\tau > y_\tau$ . Your formula may use any of the Boolean connectives introduced so far. Explain how you obtained your formula, and justify its correctness
Am I right in assuming that I would need to construct a formula for every single possibility where $x$ is the greater bit?