Let $C$ denote a logic block that is capable of comparing two $4$ – bit $2$’s complement numbers $A\:(a_3, a_2, a_1, a_0)$ and $B\: (b_3, b_2, b_1, b_0)$, where $a_i, b_i \in \{0, 1\}$ for $i = 0, 1, 2, 3$. The circuit $C$ has eight input lines $a_3, a_2, a_1, a_0, b_3, b_2, b_1, b_0$, and three output lines $E, L, G$ (Equal: $E$; Less than: $L$; Greater than: $G$). For example, if $A > B$, then the outputs should be $E = 0,\: L = 0$, and $G = 1$. Write the Boolean equations for the three outputs $E, \: L$, and $G$.