GATE CS 2006 Q36 — Computer Arithmetic

The carry-generate function $G$ for a multi-bit adder represents the condition under which a carry-out is generated by the addition of two numbers, independent of the external carry-in ($c = 0$).

For two 3-bit numbers $A = a_2 a_1 a_0$ and $B = b_2 b_1 b_0$, the carry-out $c_3$ from a Carry Lookahead Adder is given by:

Setting the carry-in $c = 0$, the carry-generate function $G$ is:

Using the standard definitions for bit-level generate ($g_i$) and propagate ($p_i$) functions:

• $g_i = a_i b_i$
• $p_i = a_i + b_i$

Substituting these into the expression for $G$:

Expanding the terms step-by-step:

1. First term: $a_2 b_2$
2. Second term: $(a_2 + b_2)a_1 b_1 = a_2 a_1 b_1 + a_1 b_2 b_1$
3. Third term: $$\begin{aligned} (a_2 + b_2)(a_1 + b_1)a_0 b_0 &= (a_2 a_1 + a_2 b_1 + a_1 b_2 + b_2 b_1)a_0 b_0 \\ &= a_2 a_1 a_0 b_0 + a_2 a_0 b_1 b_0 + a_1 a_0 b_2 b_0 + a_0 b_2 b_1 b_0 \end{aligned}$$

Combining all the expanded terms gives:

This matches Option A.