Mathematical Terms for Binary Operations

Question

I'm trying to represent binary operations on numbers in mathematically correct terminology.

For example given two binary numbers:

42 : 101010
13 : 001101

I want to perform 4 binary operations on them:

42 bitwise and 13 = 8  : 001000
42 bitwise or 13 = 47  : 101111
42 bitwise xor 13 = 39 : 100111
bitwise not 42 = 21    : 010101

Does the concept of an ordered set exist? Could I somehow use that with a summation to iterate across each 1/0 in 42 and 13?

Answer 1

Ordered sets exist and are often called sequences. The entire space of six-bit numbers might be written as $\{0,1\}^6$.

Even better, use $\Bbb Z_2^6$ to give you some implicitly understood operations you can exploit, with multiplication being bitwise and, and addition being bitwise xor.