Design a circuit with 3 inputs (x,y,z) representing the bits in a binary number,

Question

Design a circuit with 3 inputs (x,y,z) representing the bits in a binary number, and three outputs (a,b,c) also representing bits in a binary number. When the input is 1,2,3, the binary output should be one less than the input. When the binary input is 4,5,6 the binary output should be one greater than the input. When the input is 0, the output is 0, when the input is 7, the output is 7. Show your truth table, all computations for simplification, and the final circuit.

Please supply full answers if possible. The most important thing I need is the drawing of the final circuit. But please supply every full answer so I could check my truth table and computations! Thank you!

Explanation / Answer

You will need 3 "full adder" circuits. x, y, and z each go to an adder. Assuming z is the MSB, then when Z = 1, you need to add "001" When Z = 0, you need to add the two's complement of "001" - invert, and add 1, to get "111". So basically, a=x+1, b=y+(not z)+(carry from a), and c=z+(not_z)+(carry from b); where + means modulo 2 addition. Of course, this can be greatly simplified if you start with a truth table, and work backwards.

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