Exclusive OR (XOR)

Exclusive OR, also known as exclusive disjunction, exclusive alternation, or logical inequality, is a binary logic operator that outputs true only when the inputs are different. In symbolic logic, it is frequently represented as either (\oplus) or (\veebar).

Logical Operation

The exclusive or operation can be defined in terms of two binary inputs, A and B. The XOR operation yields:

True (or 1) if A is true and B is false, or A is false and B is true.
False (or 0) if both A and B are true or both are false.

This logical operation is fundamentally different from the logical disjunction (OR), which outputs true if at least one of its operands is true.

Truth Table

A	B	A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

Applications in Computing

XOR gates are a basic component of digital circuits and are used extensively in computer science. They are critical in the design of arithmetic logic units, cryptographic algorithms, and checksum calculations.

XOR in Cryptography

In cryptography, XOR is used due to its properties of being both reversible and non-linear. This operation is fundamental to many encryption algorithms because:

XORing a value with a key can be reversed by XORing the result with the same key, thus decrypting the data.
It provides a simple yet effective means of obfuscating data.

XOR and Binary Addition

XOR is also integral to binary arithmetic, particularly in adding two binary numbers. The XOR operation is used to determine the sum digits without carrying in binary addition:

Given two binary numbers, XOR is applied to calculate the sum digits.
An AND operation is used to determine the carry digits.

Historical Context

The concept of exclusive disjunction has been traced back to the work of Gottfried Wilhelm Leibniz. The symbol and further development of the XOR operation were later influenced by Edward Vermilye Huntington in 1904 and Claude Shannon who incorporated these logical operations in the foundation of digital circuit design.