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When we add a pair of two-bit binary numbers, say $a b$ and $c d$, we get a number of at most three bits, say $p q r$. Using standard operators of Boolean algebra, namely $\operatorname{AND}\;(\wedge), \operatorname{OR}\; (\vee)$ and $\operatorname{NOT}\;(\neg)$, derive the Boolean expressions of $p, q$ and $r$ in terms of $a, b, c$ and $d$.

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