Logic gates and logic circuits

Computer Science AS · Hardware. 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 gate takes one or two binary inputs and gives one binary output. There are six gates to know: NOT, AND, OR, NAND, NOR and XOR. A truth table lists the output for every possible combination of inputs. With n inputs there are 2ⁿ rows.

The same logic can be shown in four ways: a problem statement in words, a logic expression, a logic circuit and a truth table. You must be able to change any one into another.

Work in small stages. Find the output of each bracket or each gate first, as an extra column, then combine these columns to get the final output.

Rules to remember

NOT: the output is the opposite of the input
AND: output 1 only when both inputs are 1. OR: output 1 when at least one input is 1
NAND (NOT AND): output 0 only when both inputs are 1. NOR (NOT OR): output 1 only when both inputs are 0
XOR: output 1 when the two inputs are different
Number of rows in a truth table = 2ⁿ for n inputs: 4 rows for 2 inputs, 8 rows for 3 inputs
Evaluate brackets first; NOT applies only to the input or bracket that follows it
From words: "and" becomes AND, "or" becomes OR, "not" or "is off" becomes NOT, "exactly one of" becomes XOR

Common mistake

Students apply NOT to the wrong part. NOT (A OR B) inverts the result of the OR, but NOT A OR B inverts only A. These give different truth tables.

Worked example

In the eight-row truth table for X = (A AND NOT B) OR (B AND C), how many rows have X = 1?

A AND NOT B is 1 when A = 1 and B = 0, with C either value: rows 1 0 0 and 1 0 1.
B AND C is 1 when B = 1 and C = 1, with A either value: rows 0 1 1 and 1 1 1.
No row is in both lists, because one needs B = 0 and the other B = 1, so the OR gives 2 + 2 = 4 rows.

Answer: Four rows have X = 1.

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