Unit 3 Solutions
27
WXY’ + (W’Y’ ≡ X) + (Y ⊕ WZ)
= WXY’ + W’Y’X + (W’Y’)‘ X’ + Y (WZ)‘ + Y’WZ
= WXY’ + W’XY’ + (W + Y) X’ + Y (W’ + Z’) + Y’WZ
3.27 3.28 (a)
3.28 (b) NOT VALID. Counterexample: a = 0, b = 1, c = 0.
LHS = 0, RHS = 1. ∴ This equation is not always
valid.
3.28 (c) VALID. Starting with the right side, add consensus
terms
RHS = abc + ab’c’ + b’cd + bc’d + acd + ac’d
VALID: a’b + b’c + c’a
= a’b (c + c’) + (a + a’) b’c + (b + b’) ac’
= a’bc + a’bc’ + ab’c + a’b’c + abc’ + ab’c’
3.28 (d) VALID: LHS = xy’ + x’z + yz’
consensus terms: y’z, xz’, x’y
= xy’ + x’z + yz’ + y’z + xz’ + x’y
= y’z + xz’ + x’y = RHS
3.28 (e) NOT VALID. Counterexample: x = 0, y = 1, z = 0,
then LHS = 0, RHS = 1. ∴ This equation is not
always valid. In fact, the two sides of the equations
are complements.
LHS = (x + y) (y + z) (x + z)
= [(x + y)‘ + (y + z)‘ + (x + z)‘]‘
= (x’y’ + y’z’ + x’z’)‘ = [x’ (y’ + z’) + y’z’]‘
3.25 (f) 3.25 (g) [(a’ + d’ + b’c) (b + d + ac’)]‘ + b’c’d’ + a’c’d
= ad (b + c’) + b’d’ (a’ + c) +b’c’d’ + a’c’d
= abd + ac’d+ a’b’d’ + b’cd’ + b’c’d’ + a’c’d
c’d b’d’
= abd + a’b’d’ + b’d’ + c’d = abd + b’d’ + c’d
A’BCD + A’BC’D+ B’EF+ CDE’G+A’DEF+A’B’EF
= A’BD + B’EF + CDE’G + A’DEF (consensus)
= A’BD + B’EF + CDE’G