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What is Exclusive OR (XOR)?
Exclusive OR (XOR, EOR, or EXOR) is a special type of binary operator that is used in logic, electronics, and computer science to compare two values and determine if they are different.
A binary operator is a mathematical function that takes two input values and combines them to produce a single result. Fundamentally, XOR logic asks the question, “Are these two inputs different?” If the answer is yes, the result is 1 (true). If the answer is no, the result is 0 (false).
In classical logic, XOR may also be referred to as exclusive disjunction. In this context, the word “disjunction” means “OR” and the word “exclusive” means that only one out of two possible conditions is true.
Key Takeaways
- Exclusive OR is a special type of binary operator that compares input values.
- XOR logic can be used to implement and enforce the concept of mutual exclusivity.
- An XOR operator outputs ‘true’ if exactly one of its inputs is true and ‘false’ if both inputs are either true or false.
- In Boolean logic, true is represented by the digit ‘1’, and false is represented by the digit ‘0’.
- Even though Exclusive OR is a binary operator, it can be extended to allow more than two inputs.
How Does XOR Work?
XOR operators work by comparing two input values and using the concept of mutual exclusivity to determine the output.
This means that XOR operators will output ‘true’ if exactly one input is true and ‘false’ if both inputs are true or both inputs are false.
Truth Tables and XOR Logic
A truth table is a table used in logic, mathematics, electronics, and computer science to show all possible input values for a logical operation or circuit, along with their corresponding output values.
Truth tables can be used to provide a clear representation of how XOR logic functions. In this context, a ⊕ (circle plus) symbol is often used to represent XOR.
XOR and Boolean Logic
In Boolean logic, XOR can be used alongside other logical operators such as AND, OR, and NOT to form compound statements. This allows for the construction of complex logical conditions.
In this context, the caret symbol ^ is used to represent XOR, and the value ‘true’ is represented by the binary digit 1 (one). The value ‘false’ is represented by the binary digit 0 (zero).
Extending XOR
XOR operations can be extended (chained) to allow more than two inputs. When there is a series of XOR operations, the overall output of the entire expression will be true if an odd number of the individual operands are true.
Conversely, if an even number of operands are true, the XOR chain will output false.
The Importance of XOR Gates in Digital Systems
XOR gates are a special type of logic gate that enforces the concept of mutual exclusivity for digital systems.
In circuit diagrams, XOR gates look like OR gates, but they have a distinctive curved line on the input side to indicate mutual exclusivity.
The following properties make XOR gates versatile components in digital logic design, with applications ranging from arithmetic operations to data integrity and encryption:
XOR in Cryptography
XOR logic plays an important role in cryptography, specifically symmetric encryption.
When you use XOR to encrypt data with a key, you can easily decrypt it by applying XOR again with the same key. This works because of XOR’s self-inverse property.
Other Examples of XOR Use Cases
CPUs and GPUs use XOR logic to perform arithmetic and logical operations. For example, XOR is used in half-adders and full-adders to calculate the sum bits without carry-over.
XOR is ideal for comparing two signals to determine if they are different. That’s because when two inputs are different, the output will always be 1 (true).
XOR logic is used to generate parity bits that ensure the total number of 1s is either even or odd. This functionality is used for error detection in data transmissions.
In checksum calculations, XOR helps detect errors in data packets by combining multiple values. Since XOR has the property of “cancellation” (i.e., identical values result in 0), it can help verify data integrity.
The Bottom Line
When it comes to Exclusive OR definitions, the bottom line is that XOR operations will always output ‘true’ when one of the operands is true, but not both. This makes XOR logic ideal for identifying differences and enforcing mutual exclusivity.
