TeXipedia

oplus

Represents the direct sum operation or exclusive OR (XOR) in mathematics and logic, appearing as a circled plus symbol.

Overview

Commonly used across multiple mathematical disciplines, particularly in abstract algebra, linear algebra, and formal logic.

  • Essential in group theory for denoting direct sums of groups or modules
  • Frequently appears in computer science and digital logic as the XOR operator
  • Used in vector space theory to indicate the direct sum of subspaces
  • Appears in quantum mechanics and theoretical physics for combining independent systems
  • Often employed in mathematical proofs and formal mathematical writing where precise notation for binary operations is required

Examples

Direct sum of vector spaces V and W.

V⊕W={(v,w):v∈V,w∈W}V \oplus W = \{(v,w) : v \in V, w \in W\} 
V \oplus W = \{(v,w) : v \in V, w \in W\}

Exclusive OR (XOR) operation in boolean algebra.

A⊕B=(A∧¬B)∨(¬A∧B)A \oplus B = (A \land \neg B) \lor (\neg A \land B) 
A \oplus B = (A \land \neg B) \lor (\neg A \land B)

Direct sum of matrices in block matrix notation.

A⊕B=(A00B)A \oplus B = \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix} 
A \oplus B = \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}