Question

Draw and simplify the Karnaugh Maps of X, Y, Z for :

(a)  m0 + m1 + m5 + m7

(b) F = ∑(1, 3, 5, 4, 7)

(c)  m0 +m2 + m4 + m6

Answer 1

(a) m₀ + m₁ + m₅ + m₇

Mapping the given function in a K-map, we get 2 Pairs i.e., Pair-1 is m₀ + m₁ and Pair-2 is m₅ + m₇ 

= X’Y’Z’ + X’Y’Z + XY’Z + XYZ 

= X’Y’(Z’ + Z) + XZ(Y’ + Y)

= X’Y’ + XZ

Simplified Boolean expression for given K-map is F(X, Y, Z) = X’Y’ + XZ.

(b) F = ∑(1, 3, 5, 4, 7)

Mapping the given function in a K-map, we get 1 Pair and 1 Quad i.e., Pair is m₄ + m₅ and Quad is m₁ + m₃ + m₅ + m₇ 

= X’Y’Z + X’YZ + XY’Z + XYZ + XY’Z’ + XY’Z

= X’Z(Y’ + Y) + XZ(Y’ + Y) + XY’(Z’ + Z)

= X’Z + XZ + XY’ = Z(X’ + X) + XY’

= Z + XY’ 

Simplified Boolean expression as for given K-map is F(X, Y, Z) = Z + XY’.

(c) m₀ +m₂ + m₄ + m₆

Mapping the given function in a K-map, we get 2 Pairs i.e., Pair-1 is m₀ + m₄ and Pair-2 is m₂ + m₆

= X’Y’Z’ + XY’Z’ + X’YZ’ + XYZ’

= Y’Z’(X’ + X) + YZ’(X’ + X) = Y’Z’ + YZ’

= Z’(Y’ + Y) = Z’ 

Simplified Boolean expression for given K-map is F(X, Y, Z) = Z’.