Boolean Variables and Logic Gates

The Alphabet of Hardware

Digital circuits compute with just two voltage levels, interpreted as boolean values 1 (true, high) and 0 (false, low). Every digital operation is built from logic gates, hardware elements computing a boolean function.

NAND is universal: any boolean function can be built using only NAND gates (NOR is also universal). Gates are physical atoms; everything from an adder to a processor is built by wiring them together.

OR vs. XOR & Formula

Common pitfall: confusing OR with XOR. OR outputs 1 when at least one input is 1, including when both are 1. XOR (exclusive-OR) outputs 1 only when inputs differ ($A \neq B$).

For inputs (1, 1), OR glows (1) but XOR stays dark (0). Watch this "both are 1" edge case where they disagree.

The formal definition for XOR is:

$$A \oplus B = 1 \iff A \neq B$$

where $A$ and $B$ are boolean variables, $\oplus$ denotes XOR, and $\iff$ means "if and only if."