(A + B) → C — truth table

core8 rows, 5 true

Answer

8 rows, 5 true (0, 1, 3, 5, 7)

Why this example is worth doing

A compound antecedent, which is where operator precedence starts to matter. This tool binds implication looser than every other connective, so A + B → C without brackets parses as (A + B) → C — but not every textbook agrees, and the wrong reading gives a different table. The page states the precedence order in force, warns when an input is ambiguous enough that another convention would change the result, and shows the fully parenthesised form of what it actually parsed.

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
00001
10011
20100
30111
41000
51011
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)