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Boolean algebra is a branch of mathematics that deals with operations on logical values with binary variables. The Boolean variables are represented as binary numbers to represent truths: 1 = true and 0 = false. Elementary algebra deals with numerical operations whereas Boolean algebra deals with logical operations.

Chapter 12: Boolean Algebra

12.1 Boolean Functions

The basic operations of Boolean algebra are conjunction, disjunction, and negation. These Boolean operations are expressed with the corresponding binary operators AND, and OR and the unary operator NOT, collectively referred to as Boolean operators.

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Identities of Boolean Algebra

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12.2

Representing

Boolean Functions

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12.3 Logic Gates

Logic gates are the basic building blocks of any digital system. It is an electronic circuit having one or more than one input and only one output. The relationship between the input and the output is based on a certain logic.

A logic gate is an idealized or physical device that performs a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output.

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Half Adder

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Full Adder

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12.4

Minimization of Circuits

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Don’t Care (X) Conditions in K-Maps

For SOP and POS

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The "Don't care" condition says that we can use the blank cells of a K-map to make a group of the variables. To make a group of cells, we can use the "don't care" cells as either 0 or 1, and if required, we can also ignore that cell. We mainly use the "don't care" cell to make a large group of cells.

Don’t Care (X) Conditions in K-Maps

For SOP and POS

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Minimise the following function in SOP minimal form using K-Maps:

f = m(0,2,3,6,7,12,13,14) + d(1,4,11,15)

Explanation:

The SOP K-map for the given expression is:

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Minimise the following function in SOP minimal form using K-Maps:

f = M(5,8,9,10) + d(1,4,11,15)

Explanation:

The POS K-map for the given expression is:

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Minimise the following function in SOP minimal form using K-Maps:

f = m(1, 5, 6, 11, 12, 13, 14) + d(4)

Explanation:

The SOP K-map for the given expression is:

f = BC' + BCD' + A'C'D + AB'CD

Minimise the following function in SOP minimal form using K-Maps:

F(A, B, C, D) = m(1, 2, 6, 7, 8, 13, 14, 15) + d(0, 3, 5, 12)

Explanation:

The SOP K-map for the given expression is:

f = AC'D' + A'D + A'C + AB

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Minimise the following function in POS minimal form using K-Maps:

F(A, B, C, D) = M(6, 7, 8, 9) + d(12, 13, 14, 15)

The POS K-map for the given expression is:

Therefore, POS minimal is,

F = (A'+ C)(B' + C')

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The Quine–McCluskey

Method

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From the prime implicants, we can check the minterms involved. Like for the first one the minterms involved are (0,1,8,9), and we will put across if it appears in the minterm given in the problem. Then we need to identify by putting a circle if that minterm is appearing only once.

So the minterm involved in the second row (1,3,9,11) are not essential prime implicants. Since each of them has already appeared in the first or third row. So finally the minimized expression can be given as

Hence we have seen, how to obtain the minimized expression by using the Quine McCluskey method. The same procedure can be checked for a problem involving 5 or 6 variables.