Here's the detailed logic of the bitwise operation of the Hexadecimal Addition Calculator (http://everycircuit.com/circuit/4975984730963968).
As you can see, the circuit is divided in two parts. The upper part (the two XOR gates) represents the logic of the compared bit itself. The lower part represents the logic of the carry.
The two logic sources on top are the two compared bits (bit N). The one at the bottom is the carry of the previous bit (bit N-1). The upper LED is the result of bit N and the lower one is the carry of bit N.
Here's the truth table of this circuit :
A = Bit N of number A
B = Bit N of number B
C = Carry of bit N-1
Y = Result of bit N
Z = Carry of bit N
ABC-YZ
000-00
001-10
010-10
011-01
100-10
101-01
110-01
111-11
