Logic Gate Simulator Calculator
Free Logic gate simulator Calculator for logic & computer science. Enter values to get step-by-step solutions with formulas and graphs.
Calculator
Adjust values & calculateComplete Truth Table
| A | B | Output |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Formula
Each logic gate implements a specific Boolean operation. AND outputs 1 when all inputs are 1. OR outputs 1 when any input is 1. NAND and NOR are the complements of AND and OR. XOR outputs 1 for an odd number of 1-inputs. NOT inverts a single input.
Last reviewed: December 2025
Worked Examples
Example 1: 2-Input NAND Gate Truth Table
Example 2: 3-Input XOR for Parity Check
Background & Theory
The Logic Gate Simulator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Logic Gate Simulator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
AND: A*B | OR: A+B | NAND: (A*B)' | NOR: (A+B)' | XOR: A^B | NOT: A'
Each logic gate implements a specific Boolean operation. AND outputs 1 when all inputs are 1. OR outputs 1 when any input is 1. NAND and NOR are the complements of AND and OR. XOR outputs 1 for an odd number of 1-inputs. NOT inverts a single input.
Worked Examples
Example 1: 2-Input NAND Gate Truth Table
Problem: Determine all possible outputs for a 2-input NAND gate with inputs A and B.
Solution: NAND = NOT(AND), so output is 0 only when both inputs are 1:\nA=0, B=0: NOT(0 AND 0) = NOT(0) = 1\nA=0, B=1: NOT(0 AND 1) = NOT(0) = 1\nA=1, B=0: NOT(1 AND 0) = NOT(0) = 1\nA=1, B=1: NOT(1 AND 1) = NOT(1) = 0
Result: NAND(A,B) outputs: 1, 1, 1, 0 | Only 0 when both inputs are 1
Example 2: 3-Input XOR for Parity Check
Problem: Use a 3-input XOR gate to determine the parity of the binary value 101.
Solution: XOR outputs 1 when an odd number of inputs are 1:\nInputs: A=1, B=0, C=1\nStep 1: A XOR B = 1 XOR 0 = 1\nStep 2: (A XOR B) XOR C = 1 XOR 1 = 0\nSince output is 0, there is an even number of 1s (two 1s).\nEven parity bit = 0.
Result: XOR(1,0,1) = 0 | Even parity confirmed (two 1-bits in input)
Frequently Asked Questions
What are the basic logic gates used in digital electronics?
The seven fundamental logic gates are AND, OR, NOT, NAND, NOR, XOR, and XNOR. The AND gate outputs 1 only when all inputs are 1, while the OR gate outputs 1 when any input is 1. The NOT gate (inverter) flips a single input, outputting 0 for 1 and vice versa. NAND and NOR are the inverted versions of AND and OR respectively, and are called universal gates because any other gate can be built using only NAND or only NOR gates. XOR (exclusive OR) outputs 1 when an odd number of inputs are 1, and XNOR outputs 1 when an even number of inputs are 1. These gates form the building blocks of all digital circuits, from simple combinational logic to complex processors containing billions of transistors.
How do logic gates work at the transistor level?
Logic gates are implemented using transistors, specifically MOSFETs (Metal-Oxide-Semiconductor Field-Effect Transistors) in modern CMOS technology. A CMOS gate uses complementary pairs of NMOS (pulls output to ground for logic 0) and PMOS (pulls output to VDD for logic 1) transistors. A CMOS inverter (NOT gate) uses just 2 transistors: one NMOS and one PMOS. When the input is high, NMOS conducts pulling the output low; when the input is low, PMOS conducts pulling the output high. A 2-input NAND gate uses 4 transistors: 2 NMOS in series (both must be on to pull output low) and 2 PMOS in parallel (either can pull output high). CMOS logic consumes power primarily during switching transitions, which is why it dominates modern digital design and enables billions of transistors on a single chip.
What is the difference between combinational and sequential logic?
Combinational logic circuits produce outputs that depend solely on the current input values, with no memory of previous states. Examples include adders, multiplexers, encoders, and the basic gate combinations this simulator models. Sequential logic circuits, in contrast, have memory elements (flip-flops or latches) that store state, making outputs dependent on both current inputs and the circuit history. Examples include counters, registers, and finite state machines. A flip-flop itself is built from cross-coupled NAND or NOR gates, creating a feedback loop that maintains a stable state. Most real digital systems combine both types: combinational logic computes the next state based on inputs and current state, while sequential elements store that state. Modern processors use billions of gates combining both types to implement instruction execution pipelines.
How is XOR gate used in arithmetic and error detection circuits?
The XOR gate is uniquely important in digital arithmetic and error detection because it naturally implements binary addition without carry (modulo-2 addition). A half adder, the simplest adding circuit, uses one XOR gate for the sum bit and one AND gate for the carry bit. Full adders chain these together to add multi-bit numbers. XOR is also fundamental in parity generators and checkers: XORing all bits of a data word produces a parity bit that can detect single-bit errors during transmission or storage. In CRC (Cyclic Redundancy Check) calculations used in networking and storage, XOR operations implement polynomial division over GF(2). Additionally, XOR is used in stream cipher encryption, toggle flip-flops, Gray-to-binary code conversion, and comparison circuits that check if two bits are equal. The self-inverse property of XOR (A XOR A = 0) makes it invaluable for these applications.
What is propagation delay and why does it matter in logic circuits?
Propagation delay is the time it takes for an input change to produce a corresponding output change in a logic gate, typically measured in nanoseconds or picoseconds. It matters because it limits the maximum operating frequency of a digital circuit. If a signal must pass through N gates in series (called the critical path), the total delay is roughly N times the individual gate delay, and the clock frequency cannot exceed 1/total_delay. In modern 5nm CMOS technology, a single gate delay is approximately 10-20 picoseconds, enabling clock frequencies above 4 GHz. Different gate types have different delays: inverters are fastest (fewest transistors), followed by NAND/NOR, then AND/OR, with XOR being slowest due to more transistors. Propagation delay also causes glitches (momentary incorrect outputs) when signals arrive at different times at a gates inputs, which is why careful timing analysis is essential in high-speed digital design.
What is De Morgans theorem and how does it apply to logic gates?
De Morgans theorem states two fundamental equivalences: NOT(A AND B) = NOT(A) OR NOT(B), and NOT(A OR B) = NOT(A) AND NOT(B). In gate terms, a NAND gate is equivalent to an OR gate with inverted inputs, and a NOR gate is equivalent to an AND gate with inverted inputs. This theorem is essential for circuit optimization because it allows designers to convert between AND-OR and NAND-NOR implementations freely. When drawing circuit diagrams, De Morgans theorem is represented by moving the inversion bubble from the output to the inputs (or vice versa) while changing the gate symbol. In CMOS design, NAND and NOR are the natural gate forms, so designers use De Morgans theorem to convert arbitrary Boolean expressions into efficient NAND-only or NOR-only implementations. The theorem extends to any number of inputs and can be applied repeatedly to transform complex nested expressions.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy