A ⊕ B ⊕ C — truth table

coretrue on the 4 odd-parity rows

Answer

8 rows, 4 true (1, 2, 4, 7)

Why this example is worth doing

The output column is 1 exactly when an odd number of inputs is 1, which is the definition of odd parity and the reason this circuit appears at the transmit end of every parity-checked bus. Reading the table as a specification rather than a calculation is the skill being taught: the pattern of 1s tells you the function's name. The page links to the parity and checksum tool, where the same column is generated for eight and sixteen inputs instead of three.

Try your own input in the Truth table generator. Turn an expression into a full truth table, with a column for every sub-expression.

How the answer is reached

Truth table

Truth table — columns #, A, B, C, A ⊕ B ⊕ C
#ABCA ⊕ B ⊕ C
00000
10011
20101
30110
41001
51010
61100
71111

Compare with

Open this example in the Truth table generator

The field arrives filled in with this example’s input.

Note:

Notation this page assumes

  • Symbols: · is AND, + is OR, ⊕ is XOR, a prime or an overline is NOT. The field also takes ∧ ∨ ¬ ~ ! & | and the words.
  • Operator precedence, tightest first: NOT, then AND (including juxtaposition), then XOR/XNOR, then NAND/NOR, then OR, then IMPLIES, then IFF.
  • In a minterm index the first variable is the most significant bit, so over [A, B, C] minterm 5 is A·B̄·C.

Sources

  • Boole, An Investigation of the Laws of Thought (1854)
  • Shannon, “A Symbolic Analysis of Relay and Switching Circuits” (1938)