exclusiveOR

exclusiveOR, also known as XOR (exclusive or) or logical exclusive or, is a binary operation that returns true only when its two input operands differ. In Boolean algebra it is often denoted by the symbol ⊕ or by "!=" in programming languages. For values A and B, the result is true if exactly one of A or B is true, otherwise it is false. The truth table for exclusiveOR lists the following outcomes: when A=0 and B=0 the result is 0; when A=0 and B=1 it is 1; when A=1 and B=0 it is 1; and when A=1 and B=1 it is 0.

Everyday applications of exclusiveOR include error detection and correction schemes, such as parity checks, where the XOR of a set of bits determines if an even or odd number of ones is present. In cryptography, XOR is used to mix plaintext with a key in stream ciphers; because XOR is its own inverse, applying the same key again decrypts the data. Digital circuits use XOR gates to perform addition in binary arithmetic, particularly in half adders and full adders where the carry-out depends on both inputs using AND, while the sum uses XOR.

Mathematically, exclusiveOR is associative and commutative, behaves like addition modulo two, and blends nicely with other Boolean operations: A⊕B equals (A∧¬B)∨(¬A∧B). It also satisfies identities such as A⊕A = 0 and A⊕0 = A. The operation is fundamental in many areas of computer science, electrical engineering, and mathematics.