Consider a full-adder with the following input values : (a) x = 1, y = 0 and…

Concept:

A full adder combines three input bits — the two addend bits x, y and the incoming carry Ci — into a sum bit and a carry-out bit. The sum is the XOR of all three inputs, and the carry-out is 1 whenever at least two of the three inputs are 1 (the majority function):

• S = x XOR y XOR Ci

• Co = (x·y) + (y·Ci) + (x·Ci)

Application:

1. For x = 1, y = 0, Ci = 0: S = 1 XOR 0 XOR 0 = 1. Co = (1·0) + (0·0) + (1·0) = 0. So S = 1, Co = 0.

2. For x = 0, y = 1, Ci = 1: S = 0 XOR 1 XOR 1 = 0. Co = (0·1) + (1·1) + (0·1) = 1. So S = 0, Co = 1.

Cross-check:

An independent check uses the binary-addition identity x + y + Ci = S + 2·Co (the sum bit plus twice the carry must equal the arithmetic total of the three input bits). Case (a): 1 + 0 + 0 = 1, and S + 2·Co = 1 + 2(0) = 1 — matches. Case (b): 0 + 1 + 1 = 2, and S + 2·Co = 0 + 2(1) = 2 — matches. Both cases check out.

Therefore: for input set (a), S = 1 and Co = 0; for input set (b), S = 0 and Co = 1 — the pair given by S = 1, Co = 0 and S = 0, Co = 1.