CE2704 · Digital Logic Design
Theme 2 · Boolean algebra & logic gates

Boolean basics & the AND/OR/NOT gates

Numbers told us what a chip holds. Now: how a chip decides — the algebra of true/false, and the three gates that build everything else.

Built from first principles. Boolean algebra: George Boole, 1854.

Before you start

What you need first

  • Why digital — that every wire sits at one of two values, 0 or 1.

What you'll be able to do

  • Use 0/1 as logic values and the operators AND, OR, NOT.
  • Recognise the three gate symbols and their truth tables.
  • Evaluate an expression using the right order of operations.

What is Boolean algebra?

It's an algebra where every variable is either 0 or 1 — invented by George Boole in 1854, long before computers, to make logical reasoning precise.

Two values
0 = false / off / low / no
1 = true / on / high / yes
Three operators
AND — written · or side-by-side: A·B
OR — written +: A+B
NOT — a bar or apostrophe: A′
Heads-up: + and · here are not arithmetic addition and multiplication — they are reused symbols for OR and AND. And 0/1 are logic values, not numbers.

Variables, constants, expressions

A Boolean variable (e.g. A, ENABLE) holds 0 or 1 and can stand for a physical signal — a wire voltage, a button, a sensor flag. A constant is just the literal 0 or 1. A Boolean expression combines them with AND/OR/NOT to describe an output for every input combination:

ExpressionMeans
F = A·B + Cthree inputs, one output
F = A′·B + A·B′XOR, written out
F = A·A′always 0, whatever A is
The whole job of this theme: read, write, simplify, and reason about expressions like these — and see how a circuit implements each one.

AND — all conditions must be true

ABF
ANSI symbol — flat back, round front.

Reads: "F equals A AND B."   Notation: F = A·B.

ABF
000
010
100
111
Output is 1 only when every input is 1 — like a safety interlock that needs all conditions met.

OR — any condition is enough

ABF
ANSI symbol — curved back, pointed front.

Reads: "F equals A OR B."   Notation: F = A+B.

ABF
000
011
101
111
Output is 1 when at least one input is 1 — like an alarm that sounds if any sensor trips.

NOT — flip the value

AF
The small circle ("bubble") = inversion.

Reads: "F equals NOT A."   Notation: F = A′. Also called an inverter.

AF
01
10
The bubble shows up everywhere — it always means "invert this signal".

Combining gates: order of operations

Expressions have a precedence order, like ordinary algebra: NOT first, then AND, then OR. Use parentheses to override.

📐 Worked example

Read F = A + B·C′

$$ F = A + \big(B \cdot (C') \big) $$

NOT C first, then AND it with B, then OR A on top. The drawing of the circuit is the expression, just laid out in space.

✏️ Try it yourself

(a) Evaluate F = A·B for A=1, B=0. (b) Evaluate F = A+B for A=0, B=1. (c) Evaluate F = (A·B) + C for A=1, B=0, C=1. (d) Which gate outputs 1 only when all inputs are 1?

(a) 1·0 = 0 (AND needs all 1s). (b) 0+1 = 1 (OR needs any 1). (c) A·B = 0, then 0+C = 1 (AND binds tighter than OR). (d) the AND gate.

Recap — the whole topic on one screen

GateNotationOutput is 1 when…
ANDA·Ball inputs are 1 ("all")
ORA+Bany input is 1 ("any")
NOTA′the input is 0 ("flip")
Precedence: NOT → AND → OR. These three operators build every digital circuit in the course.

Next topic

Truth tables

We've used small truth tables already. Next we make them a tool: the complete, unambiguous way to specify any logic function — and to turn a requirement into one.

→ Truth tables