There are various Boolean identities some of which are as follows:
$A + A = A$
$A . A = A$
$A + 1 = 1$
$AB = BA$
$A + AB = A$
and the list goes on
But why doesn't there exist the following Boolean identities which to me seems correct ?
1) $A + B = 1$
2) $A \cdot B = 0$
thanks for your precious time.
Because if $A=B=0$ then $A+B=0$ which makes the first invalid. And if $A=B=1$ it makes $A\cdot B=1$ which makes the second invalid.
Your "proof" is flawed since the truth table doesn't include all combination of values for $A$ and $B$. A proper truth table is:
$$\begin{matrix} A & B & A+B & A\cdot B \\ \hline 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix}$$