A binary operator representing direct sums of vector spaces, exclusive OR in logic, or addition in cyclic groups.
Examples
Direct sum of vector spaces
V = V_1 \oplus V_2 \oplus V_3
Exclusive OR (XOR) operation in logic
A \oplus B = (A \land \neg B) \lor (\neg A \land B)
Group theory: external direct product
G = \mathbb{Z}_2 \oplus \mathbb{Z}_4 \oplus \mathbb{Z}_8