How would this digital logic problem be solved?

A switching circuit has three inputs and two outputs. The output variables, a and b, represent the first and second bits, respectively, of a binary number, N.N equals the number of inputs which are 0. For example, if x=1, y=0, z=1, then a=0, b=1. Write sum of products expansion for a and b.|||Logic Friday from

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does this for us given a truth table:



X,Y,Z,=%26gt;,b,a

0,0,0,,11

0,0,1,,01

0,1,0,,01

0,1,1,,10

1,0,0,,01

1,0,1,,10

1,1,0,,10

1,1,1,,00



the sum of products expansion is:

b = X' Y' Z' + X' Y Z + X Y' Z + X Y Z';

a = X' Y' Z' + X' Y' Z + X' Y Z' + X Y' Z';



Where ' means not. e.g. X' means NOT X.



But it could be done by hand. Just write out the truth table and then look at each line where e.g. "b" should be a one, and write a term that produces a one only for that line. So, for the first line, "b" should be one, so to make a one from 0,0,0 you need X' Y' Z' which is the first term. On the second line, "b" is not a one, so skip it. Same for the 3rd line. On the 4th line, "b" needs a one again and the table is 0,1,1 so you need X' Y Z as the second term to be added to the first. Continue for the remaining lines and repeat for "a".