Trace Odd Parity: Another Case

Track an intermediate XOR value and compare different triples with the same output.

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Watch the lesson | Practice 1: trace an odd-parity chain

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Use Boolean 0 and 1. First compute X = A XOR B, then output = X XOR C. A two-input XOR gives 1 when its inputs differ and 0 when they match. Across the three original inputs, the chain gives 1 for one or three ones and 0 for zero or two ones. X is not a fourth independent input. This is an ideal logical model, not physical wiring or an error-correcting system.

Worked example

At 001, X = 0 XOR 0 = 0, then output = 0 XOR 1 = 1. At 010, X = 0 XOR 1 = 1, then output = 1 XOR 0 = 1. Different triples can have the same parity.

Input triples 001 and 010 each contain one one and both give parity 1, although two positions differ. — Matching parity is weaker than matching original inputs. Trace each gate, then count the original ones.

Would you believe it?

Different triples can have the same odd-parity value: 001 and 010 each contain one 1. Two positions differ, yet both parity outputs are 1. Matching parity alone does not prove matching data.

Texas Instruments: three-input XOR function table

Question 1 For input triple 011, what are X and output, in that order? 1, 0 0, 1 1, 1

Question 2 For input triple 000, what is the final output? 2 1 0

Question 3 Which triple separates odd parity from a rule that accepts exactly one original one? 001 111 000

Question 4 A table tests 000 through 110 and leaves out 111. How many combinations still need testing? 8 3 1

Question 5 Triples 000 and 110 both give parity 0. What does that show? Equal parity can hide a change in two input positions. The triples must be identical. The parity bit tells us exactly which positions changed.

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