Question

Give the three representations of an XOR gate and say in words what XOR means.

Short answer

An XOR gate outputs true when an odd number of inputs are true. It is represented by its symbol, truth table, and Boolean expression.

Step-by-step solution

Identify XOR Gate

An XOR (exclusive OR) gate is a digital logic gate that outputs true or high only when the number of true or high inputs is odd. It is widely used in digital circuits.

Symbolic Representation

The symbolic representation of an XOR gate involves using the standard symbol for XOR in digital electronics, which is a curved or additional line before the OR gate symbol. It distinguishes XOR from the regular OR gate.

Truth Table Representation

Create a truth table for the XOR gate. The inputs are labeled as A and B. The output, often labeled as Q, follows these rules: \(A=0, B=0, Q=0; A=0, B=1, Q=1; A=1, B=0, Q=1; A=1, B=1, Q=0\). The truth table highlights that XOR outputs true only when exactly one input is true.

Boolean Expression Representation

The Boolean expression for an XOR gate can be written as \(Q = A \oplus B\), which reads as \(Q\) equals A XOR B. This can be expanded to \(Q = (A \land eg B) \lor (eg A \land B)\), meaning that XOR is true when either \(A\) is true and \(B\) is false, or \(A\) is false and \(B\) is true.

Key concepts

Digital Logic Gates

Digital logic gates are the fundamental building blocks of digital circuits. They process signals which represent logical values, often binary, to perform simple or complex operations. Each gate type represents a basic logical operation.

• An OR gate outputs true when at least one input is true.
• An AND gate outputs true only when all inputs are true.
• A NOT gate, or inverter, outputs the opposite logical level to its input.
• An XOR (exclusive OR) gate, a bit more specialized, outputs true only when exactly one input is true, which makes it essential in various computational algorithms and digital systems.

Digital logic gates are used to design digital systems, from computers to everyday electronics.

Truth Table

The truth table is a systematic way to express the output result of a logic gate for all possible input combinations. Each row of the table corresponds to one particular combination of inputs and the resulting output.
For an XOR gate, the truth table is straightforward:

• When both inputs A and B are 0, the output Q is 0.
• If A is 0 and B is 1, then Q is 1.
• When A is 1 and B is 0, Q is again 1.
• If both A and B are 1, the output Q becomes 0.

This table clearly shows that the XOR function generates a true output only when the inputs are different.

Boolean Expression

A Boolean expression represents the function of a digital logic gate using algebraic syntax based on logic operations. It is a mathematical notation used to formalize how logic gates operate.
For the XOR gate, the Boolean expression can be simplified to: \[ Q = A \oplus B \] This indicates that Q equals A exclusive OR B.
Another way to write this expression is:\[ Q = (A \land eg B) \lor (eg A \land B) \]This expanded form states that Q is true if A is true while B is false, or A is false while B is true. Such expressions are useful for designing and simplifying circuit logic.

Symbolic Representation

The symbolic representation in digital electronics allows us to visually discern different gate types using standardized symbols. Each logic gate has its unique shape that helps in identifying its function in a circuit.
The XOR gate symbol is distinctive: it resembles the OR gate symbol but includes an additional curved line on its input side. This differentiating arc indicates that the gate operates on a different logical premise than a standard OR gate.
These symbols are crucial when reading or designing circuit diagrams, as they aid engineers and students in understanding how a circuit works at a glance.