Digital electronics rests on Boolean logic gates that operate on binary signals. The AND gate outputs 1 only when all of its inputs are 1, expressed as \(Y = A \cdot B\). The OR gate outputs 1 when at least one input is 1, expressed as \(Y = A + B\). The NOT gate, or inverter, outputs the complement of its input, \(Y = \overline{A}\). These three gates form the basis of Boolean algebra, but additional gates simplify circuit design: NAND (the complement of AND), NOR (the complement of OR), and XOR, which outputs 1 when its inputs differ (\(Y = A \oplus B\)).
The NAND gate holds a special place in digital design because it is called a universal gate: any Boolean function, including AND, OR, and NOT, can be implemented using only NAND gates. Two fundamental identities, De Morgan's theorems, govern the manipulation of complemented expressions. The first states that the complement of a product equals the sum of the complements: \(\overline{A \cdot B} = \overline{A} + \overline{B}\). The second states that the complement of a sum equals the product of the complements: \(\overline{A + B} = \overline{A} \cdot \overline{B}\). These identities are essential tools for simplifying and transforming digital logic designs.
Beyond combinational logic, digital systems require memory. The flip-flop is a bistable circuit that stores a single bit, holding one of two stable states until triggered to change. Flip-flops are edge-triggered, responding to clock transitions, while similar level-triggered devices are called latches. Larger digital systems rely on components such as multiplexers (MUX), which select one of many input signals and route it to a single output based on select lines. Bridging the analog and digital worlds are the analog-to-digital converter (ADC), which samples a continuous signal and represents it as discrete values, and the digital-to-analog converter (DAC), which performs the reverse transformation.