Boolean Algebra

Every possible input combination, and the output for each.

1. AND: output 1 only when both inputs are 1.
2. OR: output 1 when either or both are 1.
3. NOT: one input, output is its opposite.

Shannon's connection, made physical with a battery, a lamp and two switches.

1. Wire two switches in SERIES with a lamp — the lamp lights only when both are closed. That is AND.
2. Rewire them in PARALLEL — the lamp lights when either is closed. That is OR.
3. Compare each against its truth table.

Two identities that let you trade one gate for another.

1. Build the table for NOT(A AND B).
2. Build the table for (NOT A) OR (NOT B).
3. Compare them row by row — they match.
4. Repeat for NOT(A OR B) and (NOT A) AND (NOT B).

Logic and arithmetic meet here.

1. Recall binary addition: 1+1 = 0 carry 1.
2. Note the SUM output is 1 when exactly one input is 1 — that is XOR.
3. Note the CARRY output is 1 only when both are 1 — that is AND.
4. Wire both and add two bits.

George Boole (1815-1864) was largely self-taught and became professor of mathematics at Cork. An Investigation of the Laws of Thought (1854) set out to express logical reasoning in symbols so it could be calculated rather than argued. He died at 49 of a fever, after walking to a lecture in the rain — and had no idea what his algebra would be used for.

Claude Shannon's 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits, made the connection to electrical circuits explicit and showed that any switching network could be designed and simplified algebraically. Before it, relay circuits were designed by intuition and trial; afterwards they could be derived and minimised. It is frequently called the most important master's thesis of the century, and the claim is defensible.

Boole's daughter Alice Boole Stott made original contributions to four-dimensional geometry, and another daughter, Ethel Lilian Voynich, wrote The Gadfly — the family is unusually interesting in its own right.

Where this approach reaches: every digital circuit, every database query with AND/OR/NOT, every conditional in every program, and the search syntax people use without naming it. It also underlies formal verification, where a chip design is checked exhaustively against a specification — possible precisely because the behaviour is finite and tabulable, which is the property Boole gave it.