Logic Gates, Truth Tables and Boolean Algebra
Two-State Logic: The Language of Digital Electronics
Every digital circuit, from a simple alarm to a microprocessor, is built from components that work with just two voltage levels. Logic 1 (HIGH) represents a voltage near the supply rail; Logic 0 (LOW) represents a voltage near 0 V. This two-state system is called binary logic. Because there are only two states, every circuit decision can be described mathematically using Boolean algebra — a branch of mathematics developed by George Boole that uses only 0s and 1s.
A logic gate is a small integrated circuit with one or two inputs and a single output. The output depends entirely on the combination of inputs at that moment. The relationship between inputs and output is described in a truth table and captured in a Boolean expression. You need to know five gates for your exam: NOT, AND, OR, NAND and NOR.
The NOT Gate
The NOT gate (or inverter) has one input and one output. It simply inverts the input: a Logic 1 input produces a Logic 0 output, and vice versa. The Boolean expression uses a bar (overline) to denote inversion: if the input is A, the output is \(\overline{A}\).
| A (input) | Q (output) |
|---|---|
| 0 | 1 |
| 1 | 0 |
The AND Gate
The AND gate has two (or more) inputs. The output is Logic 1 only when all inputs are Logic 1. If any input is 0, the output is 0. The Boolean expression uses a dot (or no symbol) to represent AND: \(Q = A \cdot B\) or simply \(Q = AB\).
| A | B | Q = A · B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
The OR Gate
The OR gate output is Logic 1 when at least one input is Logic 1. Only when all inputs are 0 is the output 0. The Boolean expression uses a plus sign: \(Q = A + B\).
| A | B | Q = A + B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
The NAND Gate
The NAND gate is an AND gate followed by a NOT. Its output is Logic 0 only when all inputs are Logic 1 — it is 1 in every other case. The Boolean expression is \(Q = \overline{A \cdot B}\).
| A | B | Q |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
The NAND gate is the exact inverse of AND. It is the most important gate in electronics because any logic function can be built from NAND gates alone — making it a universal gate.
The NOR Gate
The NOR gate is an OR gate followed by a NOT. Its output is Logic 1 only when all inputs are Logic 0. The Boolean expression is \(Q = \overline{A + B}\).
| A | B | Q |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
A NAND gate can act as a NOT gate if both inputs are connected together or if one input is tied permanently to Logic 1. When both inputs equal A, the truth table becomes: if A=0, output=1; if A=1, output=0 — identical to a NOT gate.
This is a frequently-examined configuration. If you see a NAND gate symbol where the two inputs share the same signal, or where one input is connected to the supply rail (Logic 1), treat it as a NOT gate. Writing its output as \(\overline{A \cdot A} = \overline{A}\) confirms the equivalence.
A common exam error: treating a NAND gate with one input at Logic 1 as if it were still a two-input NAND. When input B is permanently 1, the gate simplifies to \(Q = \overline{A \cdot 1} = \overline{A}\). The second input drops out of the expression entirely.
Writing Boolean Expressions
A Boolean expression describes the output of a logic circuit in terms of its inputs. For a circuit with three inputs A, B and C passing through an AND gate and an OR gate, the expression might be \(Q = A \cdot B + C\). Boolean algebra follows operator precedence: NOT is evaluated first, then AND (multiplication), then OR (addition). Brackets override this order just as in ordinary algebra.
For this course, expressions are kept to a maximum of three inputs. A typical sum-of-products expression for three inputs might look like \(Q = A \cdot B \cdot C + A \cdot \overline{B} \cdot C\). You also need to use Boolean identities and NAND gate redundancy to simplify logic circuits.
Boolean notation: NOT uses a bar over the variable (\(\overline{A}\)); AND uses a dot or no symbol (\(A \cdot B\) or \(AB\)); OR uses a plus sign (\(A + B\)). Operator precedence: NOT first, then AND, then OR.
A three-input truth table has \(2^3 = 8\) rows (one for every combination of 0s and 1s). Always list them in binary counting order: 000, 001, 010, 011, 100, 101, 110, 111. This guarantees you do not miss any combination.
To fill in the output column, evaluate the Boolean expression for each row by substituting the values of A, B and C. For \(Q = A \cdot B + C\): when A=1, B=0, C=1, \(Q = 1 \cdot 0 + 1 = 0 + 1 = 1\). Work methodically one row at a time; rushing is where most errors creep in.