As a first example of useful combinational logic, let’s build a device that
can add two binary digits together. We can quickly calculate what the answers
should be:

0 + 0 = 0          0 + 1 = 1          1 + 0 = 1          1 + 1 = 102

  

  So we well need two inputs (a and b) and two outputs. The low order output
 will be called Σ because it represents the sum, and the high order output
 will be called Cout because it represents the carry
 out.

  The truth table is

    

  Simplifying boolean equations or making some Karnaugh map will produce the
 same circuit shown below, but start by looking at the results. The Σ
 column is our familiar XOR gate, while the Cout column is
 the AND gate. This device is called a half-adder for reasons that will make
 sense in the next section.

    

  or in ladder logic

    

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