Logic circuits and logic expressions

Computer Science · Boolean logic. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A logic circuit is several gates joined together, where the output of one gate becomes an input of the next. A logic expression describes the same thing in writing, for example X = (A AND B) OR NOT C. You must be able to move between a problem statement, an expression, a circuit and a truth table.

From a problem statement, turn the words into operators: "and" gives AND, "or" gives OR, "not" gives NOT, "one or the other but not both" gives XOR. Use brackets to show which part is worked out first.

From a circuit, start at the inputs and write the output of each gate as you move towards the final output.

Rules to remember

Each operator in an expression needs one gate: X = (A AND B) OR NOT C needs one AND, one NOT and one OR gate.
The operation done last in the expression is the gate that produces the final output.
Brackets show what is worked out first, so that part is drawn nearest the inputs.
NOT (A OR B) is the same as A NOR B. NOT (A AND B) is the same as A NAND B.
If a condition is described as 0 (off, closed, not pressed), put NOT in front of that input.
From a truth table: for each row where X = 1, write the inputs joined by AND (with NOT on any input that is 0), then join these rows with OR.

Common mistake

Leaving out brackets changes the meaning: (A AND B) OR C is not the same as A AND (B OR C). Also check exactly what NOT applies to: NOT A OR B inverts only A, but NOT (A OR B) inverts the whole OR.

Worked example

A light L switches on (L = 1) when it is dark (D = 1) and either movement is detected (M = 1) or the manual switch is on (S = 1). Write the logic expression for L.

"Movement or the manual switch" gives M OR S.
This must be true together with darkness, so join it to D with AND.
Brackets keep the OR part together: L = D AND (M OR S).
Check: D = 0, M = 1, S = 1 gives 0 AND 1 = 0, so the light stays off in daylight, as required.

Answer: L = D AND (M OR S)

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