XOR gate Formula Explained In Detail: What is the formula for XOR gate and their functions?


Wondering what the XOR Gate Formula is? The XOR gate, also known as the exclusive OR gate, is a digital logic gate that takes two binary inputs and produces an output that is true only when the number of true inputs is odd. In other words, the output of an XOR gate is true if and only if its two inputs are different from each other.

The formula for an XOR gate can be represented using Boolean algebra as follows:

A ⊕ B = (A ∧ ¬B) ∨ (¬A ∧ B)

where A and B are the two input signals, ⊕ represents the XOR operation, ∧ represents the logical AND operation, and ¬ represents the logical NOT operation.

The truth table for an XOR gate is as follows:

ABOutput
000
011
101
110

As shown in the truth table, the output of an XOR gate is 1 (true) when the inputs are different, and 0 (false) when the inputs are the same.

The XOR gate is used in various applications, including:

  1. Addition: In digital arithmetic, an XOR gate is used to perform the exclusive OR operation in binary addition. It is one of the fundamental building blocks for constructing adders and subtractors.
  2. Error detection: XOR gates are commonly used in error detection and correction circuits. By comparing the input and output signals, errors can be detected when the XOR output is non-zero.
  3. Data transmission: XOR gates are used extensively in communication systems for data transmission and encoding. They are used to perform operations such as parity checking to ensure the integrity of transmitted data.
  4. Logic functions: The XOR gate can be used to implement other logic functions, such as the XNOR gate (which is the complement of XOR) and various combinational functions.

In digital circuit design, an XOR gate can be constructed using different combinations of basic logic gates like NAND gates, NOR gates, or a combination of both. This flexibility allows designers to implement XOR gates using various technologies and optimize for specific requirements.

The XOR (exclusive OR) gate is a fundamental digital logic gate that produces a true (1) output when the number of true (1) inputs is odd. The formula for a 2-input XOR gate can be expressed using standard logic notation as follows:

XOR(A, B) = A · B’ + A’ · B

Now, let’s explain this formula in detail:

  1. XOR(A, B): This represents the output of the XOR gate when provided with two inputs, A and B.
  2. A · B’: This part of the formula represents the product (AND operation) of input A and the complement (NOT) of B. In other words, it is true when A is true (1) and B is false (0).
  3. A’ · B: This part of the formula represents the product of the complement of A and B. It is true when A is false (0) and B is true (1).

The XOR gate, by definition, produces a true output when the number of true inputs is odd. So, the formula reflects this behavior. If either A is true and B is false (A · B’) or A is false and B is true (A’ · B), the output of the XOR gate will be true (1). If both A and B are true or both are false, the output is false (0).

Here’s the truth table for a 2-input XOR gate that shows the relationship between the inputs A and B and the output:

| A | B | XOR(A, B) |
|---|---|----------|
| 0 | 0 |    0     |
| 0 | 1 |    1     |
| 1 | 0 |    1     |
| 1 | 1 |    0     |

This truth table illustrates that the XOR gate outputs 1 when the inputs are different and 0 when the inputs are the same, which aligns with the formula XOR(A, B) = A · B’ + A’ · B.

Eddie Mcfarren

Eddie Is no stranger to technical writing after spending years in Networking, IT Infrastructure management, and online content marketing. He is an avid researcher, Software and apps dev tester who spends hours solving problems behind the scenes. Get in touch with him via social media and you can email him via contact@gawkygeek.com

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