Question

Expand the following Boolean functions into their canonical form:
i. f(X,Y,Z)=XY+YZ+ X Z+ X Y
ii. f(X,Y,Z)=XY+ X Y + X YZ

Answer 1

Answer on Question #43499, Engineering, Other

Expand the following Boolean functions into their canonical form:

i. $f(X,Y,Z) = XY + YZ + XZ + XY$

ii. $f(X,Y,Z) = XY + XY + XYZ$

Solution.

$$\begin{array}{l} XY + YZ + X'Z + X'Y \\ = XY(Z + Z') + (X + X')YZ + X'(Y + Y')Z + X'Y(Z + Z') \\ = XYZ + XYZ' + XYZ + X'YZ + X'YZ + X'Y'Z + X'YZ \\ + X'YZ' = XYZ + XYZ' + X'YZ + X'Y'Z + X'YZ' \\ = m_1 + m_2 + m_3 + m_6 + m_7 \\ \end{array}$$

Answer: $XY + YZ + X'Z + X'Y = m_1 + m_2 + m_3 + m_6 + m_7$

$$\begin{array}{l} XY + X'Y + X'YZ = XY(Z + Z') + X'Y(Z + Z') + X'YZ \\ = XYZ + XYZ' + X'YZ + X'YZ' + X'YZ \\ = XYZ + XYZ' + X'YZ + X'YZ' = m_2 + m_3 + m_6 + m_7 \\ \end{array}$$

Answer: $XY + X'Y + X'YZ = m_2 + m_3 + m_6 + m_7$