Digital logic and Boolean algebra

Digital logic is the branch of electronics concerned with circuits that operate on discrete signals, usually represented as two states written as 0 and 1, or low and high voltage. The basic building blocks are logic gates: AND, OR and NOT, together with the derived gates NAND, NOR, exclusive-OR (XOR) and exclusive-NOR (XNOR). Each gate produces an output that depends only on the present combination of its inputs, behaviour that can be described completely by a truth table.

The mathematics behind these circuits is Boolean algebra, formalised by George Boole. It treats variables that take only the values true and false and defines operations corresponding to AND, OR and NOT. A small set of identities — including the commutative, associative and distributive laws — lets a designer manipulate logic expressions. De Morgan's laws are particularly useful: they state that the complement of an AND equals the OR of the complements, and the complement of an OR equals the AND of the complements. This allows any circuit to be rebuilt using only NAND or only NOR gates.

Because a function can be written in many equivalent forms, minimisation reduces the number of gates and inputs needed. The Karnaugh map is a graphical method that arranges a truth table so that adjacent cells differ in a single variable; grouping neighbouring 1s lets common terms be cancelled by inspection. Larger problems use algebraic methods or computer tools. Combinational logic built this way underlies adders, multiplexers, decoders and the address logic of memory devices such as random-access memory.

Frequently asked questions

What are De Morgan's laws used for?

They convert between AND and OR forms by complementing terms, which lets a circuit be implemented entirely with NAND or NOR gates and helps simplify expressions.

What does a Karnaugh map do?

It is a visual way to minimise a Boolean function. Adjacent cells differ by one variable, so grouping 1s reveals terms that can be combined, reducing the gate count.

Why are NAND and NOR called universal gates?

Any logic function can be built from NAND gates alone or NOR gates alone, so a single gate type can implement an entire design.