A + 1 = 1 — Boolean algebra laws

edge caseannulment / null element

Answer

verified equivalent (both sides → 1)

Why this example is worth doing

The annulment law, paired on the page with its dual A·0 = 0 and carefully distinguished from the identity laws A + 0 = A and A·1 = A. Four short identities, and mixing them up is the most common source of a derivation that goes wrong in its second line. The page sets all four in one table with the operator and the constant as the two axes, which makes the pattern — the constant that matches the operator's dominant element wins — visible instead of memorised.

Try your own input in the Boolean laws & theorems. Every law, both duals, each with a truth-table proof you can check.

How the answer is reached

The law

A + 11Both sides have the same truth table over the union of their variables.

Truth table

Truth table — columns #, A, F
#AF
001
111

Compare with

Open the Boolean laws & theorems

This input is entered in the tool itself — it is too rich for a link to carry.

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)