Logical Behavior of an XOR Gate

An XOR gate outputs \(\mathsf{1}\) only when odd number of inputs is \(\mathsf{1}\). In the remaining cases (when even number of inputs is \(\mathsf{1}\)), the output is \(\mathsf{0}\). An XOR gate has a minimum of \(\mathsf{2}\) inputs and \(\mathsf{1}\) output signal.

Extended and Unambiguous Definition

Usually, when students are introduced to XOR gates, they’re taught that these gates perform an exclusive OR operation—producing a high output (\(\mathsf{1}\)) only when exactly one of the inputs is high. This interpretation works well for two-input gates. However, when constructing truth tables for XOR gates with three or more inputs, this definition breaks down. In such cases, a broader rule is introduced: the XOR gate outputs 1 when an odd number of input bits are high—effectively encoding it as an odd parity gate.

This shift often requires students to reevaluate their understanding of XOR behavior, which can lead to confusion. To eliminate this ambiguity, we adopt a unified definition: an XOR gate produces a high output when an odd number of inputs are high, regardless of how many inputs the gate has.

Representation of an XOR Gate

In this section, we will explore the symbolic and mathematical representations commonly associated with the gate. Additionally, we will examine how the gate is modeled using Verilog HDL.

Symbolic Representation

The symbolic representation of all the logic gates may vary depending on regional or institutional standards. In the United States, the American National Standards Institute (ANSI) and the Military Standard (MIL-STD-806B) standardized a symbol while European organizations such as the International Electrotechnical Commission (IEC), particularly in IEC 60617, favor a rectangular symbol. The Institute of Electrical and Electronics Engineers (IEEE), through standards like IEEE Std 91-1984, closely aligns with ANSI conventions and plays a pivotal role in harmonizing gate symbols across international platforms. Other countries and standards bodies may adopt slight variations to meet local engineering norms, but the underlying function always remains universally understood.

American Symbol

The XOR gate is visually portrayed with a shape similar to the OR gate, featuring the same concave-tipped form, but distinguished by an additional curved line trailing behind its input edge. This subtle augmentation signifies its exclusive nature: on the left side, input signals approach the gate’s tapered entry, but the output only emerges from the broad right arc when an odd number of inputs is active. The doubled contour hints at a refined logical condition, where partial agreement among inputs yields a high (true) result, capturing the gate’s uniquely selective behavior.

European Symbol

In European engineering documentation, the XOR gate is represented as a rectangular block, adhering to the standardized layout used for schematic precision. Input lines enter from the left side, feeding into the gate, while a single output line exits on the right to reflect the result of the logical evaluation. Within the rectangle, the symbol “=1” is used to indicate the gate’s exclusive behavior—activating the output only when exactly one input (in a 2-input gate) line carries a high (true) signal.

IEEE Symbol

The gate’s visual representation aligns with the American ANSI style, most notably illustrated through the distinctive double-curved outline that sets it apart from the standard OR gate. This dual curvature gives the XOR symbol a layered appearance, emphasizing its exclusive logic function. Just like the AND symbol, it tends to be slightly longer than its ANSI counterpart, maintaining proportional clarity within schematic designs.

Mathematical Representation

To represent the operation of an XOR gate mathematically, we commonly use the plus symbol inside a circle, \(\mathsf{\oplus}\), denoting exclusive addition in Boolean logic. For example, the expression \(\mathsf{A\oplus{}B}\) indicates that the output is true (or high) only when exactly one of the inputs, \(\mathsf{𝖠}\) or \(\mathsf{𝖡}\), is true—capturing the gate’s selective logic. If \(\mathsf{A}\) and \(\mathsf{B}\) are two input signals and their XOR produces the output signal \(\mathsf{OUT}\), it can be written as:

\[\mathsf{OUT= A\oplus{}B} \]

If another input \(\mathsf{C}\) is added, then:

\[\mathsf{OUT= A\oplus{}B\oplus{}C} \]

Verilog HDL Representation

In Verilog HDL, the ^ symbol is used to represent the XOR operation, formerly called the Bitwise XOR operation at the dataflow and higher abstraction levels. At the gate level, the XOR gate is instantiated using the keyword xor(<output>, <inputs>). Given two input signals, A and B, the XOR gate produces an output signal OUT using the following logic (only the relevant code is shown).

assign OUT = A ^ B; // dataflow coding
xor x1 (OUT, A, B); //gate-level coding

If another input \(\mathsf{C}\) is added, then:

assign OUT = A ^ B ^ C; // dataflow coding
xor x2 (OUT, A, B, C); //gate-level coding

Truth Table of an XOR Gate

A truth table is a structured way to show the output of a logical operation or digital circuit for every possible combination of input values. It’s a staple tool in boolean algebra, digital electronics, and computer science, especially when designing or analyzing logic gates and circuits.

A truth table, typically has input columns to the left, and to the right are the derived or output expressions. Each row in a truth table represents a unque set of input combination.

Truth tables for 2-, 3-, and 4-input XOR gates are shown here.

Truth Table of a 2-Input XOR Gate

The following truth table demonstrates the behavior of a basic 2-input XOR gate, where the inputs are labeled \(\mathsf{A}\) and\(\mathsf{B}\), and the resulting output is represented as \(\mathsf{XOR}\). This gate yields a high output (\(\mathsf{1}\)) when odd number of the inputs is high; otherwise, the output remains low (\(\mathsf{0}\)). The table captures every possible combination of inputs along with their corresponding output values. The truth table is presented after the XOR gate symbol that corresponds with the truth table.

Truth Table of a 3-Input XOR Gate

The following truth table demonstrates the behavior of a basic 3-input XOR gate, where the inputs are labeled \(\mathsf{A}\), \(\mathsf{B}\) and \(\mathsf{C}\), and the resulting output is represented as \(\mathsf{XOR}\). This gate yields a high output (\(\mathsf{1}\)) when odd number of the inputs is high; otherwise, the output remains low (\(\mathsf{0}\)). The table captures every possible combination of inputs along with their corresponding output values. The truth table is presented after the XOR gate symbol that corresponds with the truth table.

Truth Table of a 4-Input XOR Gate

The following truth table demonstrates the behavior of a basic 4-input XOR gate, where the inputs are labeled \(\mathsf{A}\), \(\mathsf{B}\), \(\mathsf{C}\) and \(\mathsf{D}\), and the resulting output is represented as \(\mathsf{XOR}\). This gate yields a high output (\(\mathsf{1}\)) when odd number of the inputs is high; otherwise, the output remains low (\(\mathsf{0}\)). The table captures every possible combination of inputs along with their corresponding output values. The truth table is presented after the XOR gate symbol that corresponds with the truth table.

Can we Implement Multi-Input Gate Using 2-Input Gates in a Cascade?

A gate cascade refers to a structured arrangement where multiple logic gates are connected in series or in levels, so the output of one gate feeds into the input of another. This is especially common when you need to implement complex logic using simpler, typically 2-input gates — like building a 4-input XOR using a cascade of 2-input XOR gates. This changes the timings and other characteristics of the gate but our goal is to analyze if the logical behavior remains the same?

Implementing a 3-Input XOR Gate Using 2-Input XOR Gates

This circuit implements a 3-input XOR gate using cascaded 2-input XOR gates. Due to the associative nature of XOR logic, the behavior remains identical to a single 3-input gate, and input ordering does not affect the final output. Readers are encouraged to construct truth tables for the configuration and compare the outputs to those of a canonical 3-input XOR gate. This verification highlights that the logical behavior remains consistent, regardless of the gate arrangement or input order.

Implementing a 4-Input XOR Gate Using 2-Input XOR Gates

This illustration demonstrates a 4-input XOR gate constructed using cascaded 2-input XOR gates. Two implementations are provided: one with two levels of cascading, and another using three distinct levels of 2-input gates. Because XOR operations are associative, both versions produce logically equivalent outputs regardless of input order. Readers are encouraged to generate truth tables based on each configuration and compare them against the canonical 4-input XOR gate to confirm identical logical behavior.

Basic Quiz about XOR Gates