Algebra for bitwise operators

Question

I am trying to prove that a certain equation, with an operation defined for base 10, is equivalent to another operation (with slightly different numbers) that is only defined in base 2 (&, |, etc.). I am trying to prove this through induction, but obviously need algebra to "restructure" my binary operation. My question is: what algebra is defined for bitwise operators?

Answer 1

Bitwise operators can be thought of as the regular Boolean operators acting on vectors of Boolean variables. So, 1001 & 0110 could be expressed equivalently as (true, false, false, true) VAND (false, true, true, false). Really then the operators VAND, VOR, VXOR, etc. would be specific to the space over which they operator (VAND_4, VOR_4, etc.) but even in linear algebra this distinction is not typically stressed. The identities w.r.t. VOR and VAND would be (false, false, …, false) and (true, true, …, true), respectively; the zero for VAND would be (false, false, …, false). You keep commutativity and associativity over Boolean^N this way.