3.1 Revisiting the add rules

The rules of adding is as follows (in English, sort of):

• 0 + 0 = 0 with no carry.
• 0 + 1 = 1 with no carry.
• 1 + 0 = 1 with no carry.
• 1 + 1 = 0 with a carry of 1.

This is easy for a person to understand, but it does not help us understand how a (binary) processor performs the operations. Let us define a function C(x,y) that returns the carry (1 or 0) to the next digit, given that we are adding x and y (both are either 0 or 1). We also define a function R(x,y) that returns the result (1 or 0) of adding x and y.

Using this notation, we can then define:

• C(0,0) = 0,R(0,0) = 0
• C(0,1) = 0,R(0,1) = 1
• C(1,0) = 0,R(1,0) = 1
• C(1,1) = 1,R(1,1) = 0

Can we implement this in “logic” gates? The most commonly known logic gates are “and” (conjunction) and “or” (disjunction). We already know these constructs as logical operators in C/C++ and other high level programming languages.