Trace an Odd-Parity Chain

Follow two XOR gates and distinguish an odd count from exactly one.

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Each input is 0 or 1. Two-input XOR gives 1 for different values and 0 for matching values. First compute X = A XOR B, then output = X XOR C. For three original inputs, this output is 1 for an odd number of ones: one or three. It is 0 for zero or two ones. This is an ideal logical model, not physical wiring or a guarantee about message accuracy.

Worked example

For A = 0, B = 1, C = 1: X = 0 XOR 1 = 1. Then output = 1 XOR 1 = 0. Two original inputs are 1, an even count.

For input triple 110, the first XOR gives X = 0 and the second XOR with C = 0 gives output 0. — Work through each gate. Keep the intermediate X separate from the final output.

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 101, what are X and output, in that order? 0, 1 1, 1 1, 0

Question 2 For input triple 100, what is the final output? 0 1 2

Question 3 Why does input triple 111 give output 1? Exactly one input is 1. The gate adds them to make 3. Three ones are an odd count.

Question 4 How many rows test every combination of three Boolean inputs? 3 6 8

Question 5 Triples 001 and 010 both give parity 1. What can you conclude? The original triples must be identical. Matching parity does not prove matching triples. Parity identifies the exact changed position.

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