Digital Logic

Digital Logic is the foundation of all computing. At the physical level, computers are just vast networks of Transistor(s) that implement Boolean Logic

Logic Gates

A Gate is a physical device that implements a Boolean function.

• 0 (Low/No Voltage): False.
• 1 (High Voltage): True.

Gate	Symbol	Mathematic Symbol	Output
AND	$A\cdot B$	$\wedge$	True if and only if both inputs are 1.
OR	$A+B$	$\vee$	True if at least one input is 1.
NOT	$\bar{A}$	$\neg$	True if input is 0. (Inverts Input).
XOR	$A\oplus B$	$\oplus$	True if inputs are different.
NOR			True if and only if both inputs are 0.
NAND	$\overline{A\cdot B}$	$\neg (\wedge )$	True unless both inputs are 1.

Universal Gates: NAND and NOR are “Universal.” You can build any other gate (AND, OR, NOT) using only NAND gates. This is crucial for manufacturing Central Processing Unit(s) (CPU) cheaply.

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Combinational Logic

Circuits where the output depends only on the current inputs (no memory).

• Adder: A circuit that adds binary numbers.
  • Half Adder: Adds 2 bits (Sum + Carry). Uses XOR and AND.
  • Full Adder: Adds 3 bits (A, B, Carry-In).

Sequential Logic

Circuits that have Memory (State). The output depends on inputs and the previous state.

• Flip-Flop: A circuit that stores 1 bit of data.
• Clock: A signal that oscillates (0 → 1 → 0) to synchronize state updates.
• Registers: A collection of Flip-Flops used to store a “Word” (e.g., 64 bits).

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From Logic to C

When you write if (a && b) or use bitwise operators like a & b (see Bitwise Operations), the Central Processing Unit(s) (CPU) executes instructions that physically route electrons through a series of AND gates in the ALU.

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