L5 boolean theorems, Lecture notes of Digital Electronics

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15EC304 Digital System

Design

Lecture-5 Boolean Laws and Theorems

Session Objectives

1. Demonstrate the ability to use basic Boolean laws.

2. Use the sum-of-products method to design a logic circuit

based on a design truth table.

Basic Laws

1) The commutative laws are

A + B = B + A

AB = BA

These two equations indicate that the order of a logical

operation is unimportant because the same answer is arrived

at either way

Basic Laws

2) The associative laws are

A + (B + C) = (A + B) + C

A(BC) = (AB)C

These laws show that the order of combining variables has no

effect on the final answer.

3) The distributive law is

A(B + C) = AB + AC

This law is easy to remember because it is identical to

ordinary algebra

OR Operations A + 0 = A A + A = A A + 1 = 1 A + A’ = 1 AND Operations

A. 1 = A

A. A = A

A. 0 = 0

A. A’ = 0

Double Inversion and De Morgan’s Theorems

The double-inversion rule is

(A’)’ = A

De Morgan theorems

(A + B)’ = A’.B’

(A.B)’ = A’ + B’

Dual and Self-Dual The dual of a switching function is one generated if (i) ‘+’ is replaced with ‘. ’ , (ii) ‘. ’ is replaced with ‘+’ and (iii) ‘1’ or ‘0’ if appear are complemented. F(A, B) = A. B’ + A’ .B Its dual will be FD (A, B) = (A + B’).(A’ + B) = A. A’ + A.B + B’.A’ + B’ .B = A.B + A’. B’ The dual of a self-dual function is the function itself.

Dual and Self-Dual The dual of a self-dual function is the function itself. F(A, B, C) = A.B + B.C + C.A (2.22) Its dual function, FD(A, B, C) = (A + B).(B + C).(C + A) (2.23) Let us simplify using basic laws. FD(A, B, C) = (A + B).(B + C).(C + A) = (A.B + C.A + B.B + B.C).(C + A) = (A.B + C.A + B + B.C).(C + A) = A.B.C + C.A + B.C + B.C + A.B + C.A.A + B.C.A + B.C.A = A.B.C + C.A + B.C + B.C + A.B + C.A + A.B.C + A.B = A.B.C + A.B + B.C + C.A =A.B.1 + B.C + C.A = A.B + B.C + C.A

Covering and Combination

The covering rule, where one term covers the condition of the

other term so that the other term becomes redundant, can be

represented in dual form as

A + AB = A and

A(A + B) = A

Consensus Theorem  (^) The consensus theorem finds a redundant term which is a consensus of two other terms.  (^) The idea is that if the consensus term is true, then any of the other two terms is true and thus it becomes redundant. This can be expressed in dual form as A.B + A’.C + B.C = A.B + A’.C (A + B) ( A’ + C) (B + C) = (A + B) ( A’ + C)

Self Test

1. All the rules for Boolean algebra are exactly the same as for ordinary algebra. (T or F)
2. Expand using the distributive law: Y = A(B + C).
3. Simplify: Y = A Q + AQ. Upcoming

Sum of Products methods/Karnaugh map-Introduction