Boolean Algebra
     
      Binary addition
      The half adder
      The Binary addition circuit - the full adder
     
Binary addition
The binary system works under the same principles as the decimal system as show basic rules for binary addition:
(1) 0 + 0 = 0
(2) 0 + 1 = 1
(3) 1 + 0 = 1
(4) 1 + 1 = 1 0 (with a carry of 1)
(5) 1 + 1 + 1 = 1 1 (with a carry of 1)
or
(1) (2) and (3)    (4)    (5)
           1
0        1      1     1
+ 0    +  0  +  1 +  1
0        1    10   11
 
 Example:
1 1    carry
1 1 0 1 0 =>    26
+ 1 1 1 0 0 => + 28
1 1 0 1 1 0   sum    54
When adding two multiple digits numbers a carry has to be added to the next higher place value digit.
Thus, in order to design a logical circuit that performs a binary addition we should form a truth table with two columns for binary inputs,
a and b, and columns for the outputs, the sum (S) and carry (c).
The half adder
  a b a b a b a b carry Sum
1 0 0 1 1 0 0 0 0
2 1 0 0 1 0 1 0 1
3 0 1 1 0 1 0 0 1
4 1 1 0 0 0 0 1 0
The above circuit is named a half adder because it only register a carry as the result of addition of the two binary 1's from the input (marked with 4).

The circuit which is capable to perform the addition of three bits (as the column marked with 6 in the right example), i.e., that includes a carry from a previous column, is called a full adder and is shown below.
    64 32 16 8 4 2 1  
  8 7 6 5 4 3 2 1
  0 0 1 1 0 1 1 0        a = 54
  0 0 1 0 1 1 0 1        b = 45
0 0 1 1 1 1 0 0     +   c (carry)
  0 1 1 0 0 0 1 1        S = 99
The Binary addition circuit - the full adder
  
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