Express the following function in a sum of minterms and a product of maxterms

Express the following function in a sum of minterms and a product of maxterms:

F(A,B,C,D) = D(CA’) + B’D

The correct answer and explanation is :

To express the function ( F(A,B,C,D) = D(CA’) + B’D ) in both a sum of minterms (SOP) and a product of maxterms (POS), follow these steps:

Step 1: Expand the Given Function

The given Boolean function is:

[
F(A,B,C,D) = D(CA’) + B’D
]

Distribute Terms

Expanding using the distributive property:

[
F(A,B,C,D) = DCA’ + B’D
]

Factor (D) in the first term:

[
F(A,B,C,D) = D(A’C + B’)
]

Step 2: Find the Sum of Minterms (SOP)

A minterm is a product (AND) of all variables in either complemented or uncomplemented form.

Identify Minterms

We analyze each product term:

1. (DCA’):

• ( A = 0, B = -) (does not appear), ( C = 1, D = 1 )
• Possible values for ( B ): 0 or 1
• Corresponding minterms: ( (0,0,1,1) = m3 ) and ( (0,1,1,1) = m7 )

1. (B’D):

• ( B = 0, D = 1 ) (A and C are free)
• Possible values: ( A,C = 00, 01, 10, 11 )
• Corresponding minterms: ( m1, m3, m5, m7 )

Combine the minterms

[
F(A,B,C,D) = \sum m(1,3,5,7)
]

Step 3: Find the Product of Maxterms (POS)

A maxterm is a sum (OR) of all variables in either complemented or uncomplemented form.

Find the Missing Minterms

The function covers minterms ( 1,3,5,7 ), so the missing minterms (for a 4-variable function) are:

[
M_0, M_2, M_4, M_6, M_8, M_9, M_{10}, M_{11}, M_{12}, M_{13}, M_{14}, M_{15}
]

Convert to Maxterms

[
F(A,B,C,D) = \prod M(0,2,4,6,8,9,10,11,12,13,14,15)
]

Final Answer

1. Sum of minterms (SOP):
   [
   F(A,B,C,D) = \sum m(1,3,5,7)
   ]
2. Product of maxterms (POS):
   [
   F(A,B,C,D) = \prod M(0,2,4,6,8,9,10,11,12,13,14,15)
   ]

Explanation

• The sum of minterms form represents the function as an OR of all the minterms where ( F = 1 ).
• The product of maxterms form represents the function as an AND of all the maxterms where ( F = 0 ).
• The process involves expanding the function into product terms, identifying the minterms it covers, and finding the missing maxterms to construct the complementary representation.

This method ensures the function is correctly expressed in both forms.