Learning Objectives:

1. Identify minterms (product terms) and maxterms (sum terms).
2. List the standard forms for expressing a logic function and give an example of Sum Of Products (SoP)

• The Sum-of-Products (SOP) form is a method (or form) of simplyfying the Boolean expression of logic gates.
• Sum and product derived from the symbolic representations of the OR and AND functions.
• OR (+), AND(.), addition and multiplication f(A,B,C) = ABC + A'BC'

Solved Example: 9981-01

The minimised form of Boolean logic expression (A’B’C’ + A’BC’ + A’BC + ABC’) can be reduced to: (ISRO (VSSC) Technical Assistant Electrical 2016)

A. A’C’ + BC’ + A’B

B. A’C’ + B’C’ + A’B

C. A’C + BC + A’B

D. AC + BC’ + AB

Correct Answer: A

Solved Example: 9981-02

If the Boolean expression P̅Q + QR + PR is minimized, the expression becomes: (ESE Electronics 2011 Paper II)

A. $\bar{P}Q + QR$

B. $\bar{P}Q + PR$

C. $QR + PR$

D. $\bar{P}Q + QR + PR$

\begin{align*} \bar{P}Q + QR + PR &= \bar{P}Q + PR + QR (\bar{P} + P)\\ &= \bar{P}Q + PR + QR\bar{P} + QRP\\ &= \bar{P}Q(1 + R) + PR(1 + Q)\\ &= \bar{P}Q + PR \end{align*}

Correct Answer: B

• When two or more sum terms are multiplied by a Boolean OR operation.
• Sum terms are defined by using OR operation and the product term is defined by using AND operation.
• f(A,B,C) = (A' + B) . (B + C')

Learning Objectives:

1. Use a K-map to minimize a logic function (including those that are incompletely specified) and express it in either minimal SoP or PoS form.

• The Karnaugh map, also known as the K-map, is a method to simplify Boolean algebraic expressions.
• The K-map reduces the need for extensive calculations by taking advantage of human's pattern recognition capability.
• The required Boolean results are transferred from a truth table onto a two -dimensional grid where the cells are ordered in Grey code, and each cell position represents one combination of input conditions, while each cell valuerepresents the corresponsing output value. Optimal groups of one's and zero's are identified.
• These terms can be used to write a minimal Boolean expression representing the required logic.

• Groups may not include any cell containing a zero.
• Groups may be horizontal or vertical but NOT diagonal.
• Groups muct contain 1,2,4,8 or in general 2$^n$ That is if n= 1, a group will contin two one's and if n = 3, the group will contain four one's
• Each group should be as large as possible.
• Each group containing a one must be at least in one group.
• Groups may overlap.
• Groups may wrap around the table.
• There shoul dbe as few groups as possible, as long as this does not contradict any of the previous rules.

Solved Example: 9956-01

The simplification in minimal sum of product (SOP) of \[Y = F(A, B, C, D) = \sum_m(0, 2, 3, 6, 7) + \sum_d(8, 10, 11, 15)\] using K-maps is: (ESE Electronics 2017)

A. Y = AC + BD̅

B. Y = AC̅ + BD̅

C. Y = A̅ C̅ + B̅ D

D. Y = A̅ C + B̅ D̅

Correct Answer: D

Solved Example: 9956-02

Simplified expression/s for following Boolean function F(A, B, C, D) = ∑ (0, 1, 2, 3, 6, 12, 13, 14, 15) is/are: (UGC NET CS 2020)

A. A'B' + AB + A'C'D'

B. A'B' + AB + A'CD'

C. A'B' + AB + BC'D'

D. A'B' + AB + BCD'

Correct Answer: D

Solved Example: 9956-03

A K-map of 3 variables contains _______ cells. (UPPCL JE Nov 2019 Shift II)

A. 8

B. 3

C. 6

D. 9

Correct Answer: A

Solved Example: 9956-04

How many variables do 16 squares eliminate? (DFCCIL Executive S&T Sept 2021)

A. 7

B. 4

C. 11

D. 1

Correct Answer: B