Proving the Boolean expressions

Question

Are these two Boolean expressions the same?

*$co$ is the carry out while $ci$ is the carry in.

Answer 1

Since $a+b = a \oplus b +ab$, we have \begin{eqnarray} c_{o} &=& ab+c_{in} (a+b) \\ &=& ab+c_{in} (a \oplus b+ab) \\ &=& ab+c_{in} a \oplus b+c_{in} ab \\ &=& ab+c_{in} a \oplus b \end{eqnarray} (Since $x+yx = x$.)