Electronic – Theory for 3-bit full adder and calculations with un-/signed integers

The standard adder / subtract circuit is given by the figure below.For an explanation of how it works please see my previous answer here : Subtraction using adder circuit

Now notice that if you were to change the input that is tied to '0' to Cin, then the circuit would now perform the operation: A + ~B + (Cin ^ SUB), meaning that:

1) it would be a normal Adder/Subtract circuit when Cin is 0.

2) if Cin = 1 and Sub = 1 it would evaluate to:

A + ~B + (1 ^ 1) = A + ~B = A + ~B + 1 - 1 = A + (~B + 1) - 1 = A - B - 1 ,

This means that the normal Add/Sub circuit can achieve the operation you require, all you need to do is change the input tied to '0' your Cin.

You could also design the desired circuit by using the usual method of writing all your possible inputs into a truth table, then deriving the Boolean equations for them, then simplifying them to get your final circuit equation.

The blocks labeled "FA" are called full adders, and they are fundamental to how binary arithmetic is done at the gate level. The circuit you have presented here is an implementation of a 4-bit adder/subtractor. Combining addition and subtraction in the same operation requires the use of an alternative representation of binary numbers. This is called 2's complement.

It turns out that doing the operation $$A - B = C$$ is the same as doing the operation $$\overline{A} + 1 + B = C$$ The A with a line over it means we have taken the complement of A, which simply means we have swapped 0's with 1's and 1's with 0's. Example: $$\overline{1100010} + 1 = 0011101 + 0000001 = 0011110$$ In the circuit you provided, the XOR-gates perform the function of flipping the input bits A if S = 1. The input S also provide an additional carry bit into the first full adder, which represents adding 1 to the number. The short answer to your question is this: $$ A = 0100, B = 0111$$ $$A - B = 0100 - 0111 = 0100 + (\overline{0111} + 1) = 0100 + 1001 = 1101$$

Our result looks fishy, but note that when we put a 2's complement in, we'll get one out. This is the digital representation of the negative number, and by taking the complement again, we can more easily see the actual value. $$\overline{1101} = 0010 + 0001 = 0011$$ Which matches your prediction.

If you're looking to understand digital logic and binary arithmetic, I would advise you to not think of S = 1 as simply meaning subtraction. In practice, the function of S is to determine whether or not a subtraction should take place, but it won't give any insight into why it does that. Look up a cuircuit diagram of a single full adder, understand how it works and then go back the problem of a ripple carry adder/subtractor.