From English to a Boolean expression

The everyday engineering skill: turn a written requirement — a safety rule, a warning condition — into precise logic a machine can follow exactly.

Built from first principles.

Before you start

What you need first

Truth tables — to verify a translation, row by row.
Theorems & DeMorgan — to check two phrasings agree.

What you'll be able to do

Follow a repeatable method from requirement to expression.
Translate connectors (and / or / not / only if).
Handle negatives correctly — the #1 source of bugs.

A repeatable method for any requirement

List the signals. Name every condition and decide which value (0/1) means "active" — your signal convention.
Identify the output. What single yes/no decision are we making? Name it.
Translate the connectors. "and"→AND, "or"→OR, "not / unless / without"→NOT, "but only if"→AND, "either…or"→OR.
Watch the negatives. "Not pressed" means you want the signal to be 0 → use X′, not X.
Write the truth table and check it against the spec — every row.

This scales from 2 inputs to 20. The most common bug is forgetting one row — exactly the kind of bug that hurts people in safety systems.

📐 Worked example

Motor-drive interlock

Spec: "Energize the motor if the start button is pressed AND the emergency-stop is NOT pressed AND the inverter reports ready."

Signals: START (1 = pressed), ESTOP (1 = pressed), READY (1 = ready); output RUN (1 = energize).

$$ \mathrm{RUN} = \mathrm{START}\cdot \mathrm{ESTOP}'\cdot \mathrm{READY} $$

START	ESTOP	READY	RUN
1	0	1	1

Only this row gives RUN = 1; every other combination gives 0. Pressing the e-stop forces RUN to 0 — exactly what safety needs. Note ESTOP′: "NOT pressed."

📐 Worked example

Seatbelt warning — and the same rule written two ways

Spec: "The warning light turns on when the ignition is on, the seat is occupied, AND the seatbelt is unbuckled."

Signals: IG (1 = on), S (1 = occupied), B (1 = buckled); output W (1 = light on). "Unbuckled" = B′:

$$ W = \mathrm{IG}\cdot S \cdot B' $$

The safety team rephrases from the opposite side: "the warning is OFF when ignition is off, OR the seat is empty, OR the belt is buckled." That describes W′:

$$ W' = \mathrm{IG}' + S' + B $$

Negate both sides and apply DeMorgan — and you get the same expression back:

$$ W = (\mathrm{IG}' + S' + B)' = \mathrm{IG}\cdot S \cdot B' $$

DeMorgan proves the two phrasings are the same circuit — one suits technicians ("when does it light?"), the other suits safety reviewers ("when is it off?").

Common pitfalls

The missing inversion. "Light on when belt unbuckled" needs B′, not B. Forgetting it is the most common safety-logic bug.
The forgotten row. Always finish with the truth table and check every combination — silent rows are still part of your design.
"Only if" vs "if". "Run only if ready" makes READY a required AND term; don't drop it.

✏️ Try it yourself

Translate: "A greenhouse fan F should run when the temperature is HIGH (H=1) AND the window is NOT open (O=1 means open), OR when the manual override M is on." Write the expression.

"HIGH and window not open": H·O′. "…OR override on": add + M. Answer: F = H·O′ + M.

Recap — the whole topic on one screen

Step	Do this
1 Signals	name each condition + active value (convention)
2 Output	name the single yes/no decision
3 Connectors	and→·, or→+, not/unless→′, only if→·
4 Negatives	"not active" → use the complement
5 Verify	truth table, every row, against the spec

Next topic · Theme 3

Canonical forms (SOP & POS)

You can write logic from a requirement. Next: the standard way to write a function straight from its truth table — sum-of-products and product-of-sums — the launchpad for Karnaugh-map simplification.

→ Canonical forms (SOP & POS)