Addition/Subtraction

Design a system that gets four one bit inputs: a, b, cin and E, where E is the desired operation (1 – addition, 0 – subtraction). The system has two one bit outputs: sumadd / sumsub and cout / bout.

For implementing the system you have a single F.A and two xor gates.

Answer

Lets start with the truth table:

Now, obviously sumadd and cout are the outputs of the F.A. Moreover, sumadd and sumsub are equal.

We need to think of such a way to use the two xor gates and the input bit E in order to generated the desired outputs. 

We need to notice to two things:

  1. a ⊕ 0 = a and a ⊕ 1 = a’. (a ⊕ E)
  2. If the input to the F.A is a’ then we get bout at cout(output of the F.A).

The equation that describes the sum of a F.A is S = a ⊕ b ⊕ cin/bin.

Hence: 

S = E ⊕ ( E ⊕ a ⊕ b ⊕ cin/bin) = a ⊕ b ⊕ cin/bin.

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