HW3_soln

# HW3_soln - Homework#3 Solutions ECE 15a Winter 2007 1...

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Homework #3 Solutions ECE 15a Winter 2007 1. Express each of the following functions in disjunctive normal form (DNF) in the smallest possible number of variables: (a) f = (u’+v+w)(u’v+uw)’ = (u’+v+w)(u’v)’(uw)’ = (u’ +v+w)(u+v’)(u’ +w’) = [u’ + w’(w+v)] (u+v’) = (u’ + w’v)(u+v’) = u’v’ + w’vu = u’v’w + u’v’w’ + uvw’ (b) g = x’yz+(x’+y)(x’+z) = x’yz+x’+yz = x’yz+x’(y+y’)(z+z’)+(x+x’)yz = x’yz+x’yz+ x’y’z+x’yz’+ x’y’z’+xyz +x’yz = x’yz+x’y’z+x’yz’+ x’y’z’+xyz (c) h = (x’+y)(x’+y’)(x+z) = x’(x+z) = x’z 2. Write each of the following functions in DNF in the three variables x,y,z: (a) f = x’+y’ = (x’+y’)(z+z’) = x’z+x’z’+y’z+y’z’ = x’yz+x’y’z+x’yz’+x’y’z’+xy’z+x’y’z+xy’z’+x’y’z’ = x’yz+x’y’z+x’yz’+x’y’z’+xy’z+xy’z’ (b) g = xz+x’z’ = xz(y+y’)+x’z’(y+y’) = xyz+xy’z+x’yz’+x’y’z’ (3p) 3. Write the function of 3 variables x,y,z which is 1 if either x = y= 1 and z=0, or if x=z=1 and y=0; and is 0 otherwise.

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