Trace a Full Adder with Carry-In

Trace a third input without confusing carry-in with carry-out.

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Watch the lesson | Practice Sum and Carry in Signal Station

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Use bits 0 and 1. XOR gives 1 for different inputs; AND needs both ones; inclusive OR needs at least one. C is carry-in. Find X = A XOR B, Sum = X XOR C, G = A AND B, P = X AND C, then Carry = G OR P. Check A + B + C = 2 x Carry + Sum. This is an ideal one-column model, not physical wiring.

Worked example

For A = 1, B = 0, C = 1: X = 1, Sum = 0, G = 0, P = 1 and Carry = 1. Carry/Sum is binary 10, meaning two.

For A 1, B 0, carry-in C 1: X is 1, Sum is 0, G is 0, P is 1 and Carry is 1. The result binary 10 means two. — C enters this column. Carry leaves it. Their names describe different positions, not a promise they match.

Would you believe it?

A binary adder can pass one column's carry-out to the next larger column's carry-in. The carried 1 has the next place's value, just as a carried digit changes place in ordinary column addition.

MIT OpenCourseWare: full adders and carry chains

Question 1 For A = 0, B = 1, C = 1, what are X and Sum, in that order? 1, 1 0, 1 1, 0

Question 2 For A = 1, B = 1, C = 0, what is Carry-out? 1 0 2

Question 3 For A = 0, B = 0, C = 1, which Carry/Sum pair is correct? 10 01 11

Question 4 Which input triple produces Carry 1 and Sum 1? 100 110 111

Question 5 To link adjacent columns in a larger adder, where does the smaller column's carry-out go? To the next larger column's carry-in. Back into its own Sum output. It is discarded every time.

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