I just realized why the AND and OR gates map to multiplication and addition in boolean algebra. We’re essentially inputting a zero or a one into a mathematical expression.
In the case of AND, multiplication makes it so that as long as there’s a 0 in the input we’ll get a 0 in the output. This is because anything multiplied by 0 is, well, zero.
In the case of OR, addition makes it so that as long as there’s a 1 in the input, we’ll end up with a 1 in the output. This is because we have at least one 1 in the input to sum onto the result.
Ah, math is fun!