/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 Find the duals of these Boolean ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 28

Find the duals of these Boolean expressions. $$\begin{array}{ll}{\text { a) } x+y} & {\text { b) } \overline{x} \overline{y}} \\ {\text { c) } x y z+\overline{x} \overline{y} \overline{z}} & {\text { d) } x \overline{z}+x \cdot 0+\overline{x} \cdot 1}\end{array}$$

Short Answer

Expert verified
a) \( x \cdot y \) b) \( \overline{x} + \overline{y} \) c) \( ( x + y + z ) \cdot ( \overline{x} + \overline{y} + \overline{z} ) \) d) \( ( x + \overline{z} ) \cdot ( x + 1 ) + ( \overline{x} + 0 ) \)

Step by step solution

01

Understand Duality in Boolean Algebra

Duality in Boolean algebra involves swapping AND (路) with OR (+) and OR (+) with AND (路), along with swapping the constants 0 and 1.
02

Find the Dual of Expression a)

For the expression \( x + y \), apply the duality principle: AND replaces OR. Thus, the dual is \( x \cdot y \).
03

Find the Dual of Expression b)

For the expression \( \overline{x} \cdot \overline{y} \), apply the duality principle: OR replaces AND. Thus, the dual is \( \overline{x} + \overline{y} \).
04

Find the Dual of Expression c)

For the expression \( xyz + \overline{x} \overline{y} \overline{z} \), apply the duality principle: AND replaces OR and OR replaces AND. Thus, the dual is \( (x + y + z) \cdot ( \overline{x} + \overline{y} + \overline{z} ) \).
05

Find the Dual of Expression d)

For the expression \( x \overline{z} + x \cdot 0 + \overline{x} \cdot 1 \), apply the duality principle: AND replaces OR and OR replaces AND. Thus, the dual is \( ( x + \overline{z} ) \cdot ( x + 1 ) + ( \overline{x} + 0 ) \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boolean Algebra
Boolean Algebra is an integral part of digital logic and computer science. It deals with binary variables and logical operations. A binary variable can have only two possible values: 0 or 1. The primary operations used in Boolean algebra are AND (denoted as 路), OR (denoted as +), and NOT (denoted as an overline, \overline{...}). Boolean Algebra allows complex logical expressions to be simplified and manipulated using specific rules and properties.

Key properties of Boolean Algebra include:
  • Commutative Laws: \(A + B = B + A\) and \(A \cdot B = B \cdot A\)
  • Associative Laws: \(A + (B + C) = (A + B) + C\) and \(A \cdot (B \cdot C) = (A \cdot B) \cdot C\)
  • Distributive Laws: \(A \cdot (B + C) = (A \cdot B) + (A \cdot C)\)
  • Identity Laws: \(A + 0 = A\) and \(A \cdot 1 = A\)
These properties allow for the systematic simplification of Boolean expressions, making it easier to design and analyze logical circuits.
Duality Principle
The Duality Principle is a cornerstone of Boolean Algebra. It states that every Boolean equation has a dual. To obtain the dual of a Boolean expression, you swap AND (\cdot) with OR (+) and 0 with 1, and vice versa. This principle helps in understanding the inherent symmetry in Boolean algebraic structures.

Let's look at some simple examples:
  • Original: \(A \cdot B + 0 = A \cdot B\)
    Dual: \(A + B \cdot 1 = A + B\)
  • Original: \(A + B = B + A\)
    Dual: \(A \cdot B = B \cdot A\)
The duality principle is useful not just for theoretical understanding but also for practical simplification tasks in digital circuit design.

As demonstrated, applying the dual involves performing the specified swaps on every part of the expression. This includes complex expressions, making it easy to verify Boolean identities.
Boolean Expressions
Boolean Expressions are mathematical expressions used to represent logical statements. They can include variables, constants (0 and 1), and logical operations (AND, OR, NOT). Simplifying Boolean expressions makes it easier to design efficient digital systems.

Consider the following steps to find the dual of some Boolean expressions:
  • Expression a: Given \(x + y\), its dual is found by replacing + with \cdot. The dual is \(x \cdot y\).
  • Expression b: Given \overline{x} \cdot \overline{y}, its dual is found by replacing \cdot with +. The dual is \overline{x} + \overline{y}.
  • Expression c: Given \(xyz + \overline{x} \overline{y} \overline{z}\), its dual is found by swapping AND with OR and vice versa. The dual is \( (x + y + z) \cdot (\overline{x} + \overline{y} + \overline{z})\).
  • Expression d: Given \(x \overline{z} + x \cdot 0 + \overline{x} \cdot 1\), its dual is found by making the specified swaps. The dual is \(( x + \overline{z} ) \cdot ( x + 1 ) + ( \overline{x} + 0)\).
Understanding how to manipulate and find the dual of Boolean expressions aids in optimizing logic circuits and verifying logical statements.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the sum-of-products expansion of the Boolean function \(F(w, x, y, z)\) that has the value 1 if and only if an odd number of \(w, x, y,\) and \(z\) have the value \(1 .\)

Another way to find a Boolean expression that represents a Boolean function is to form a Boolean product of Boolean sums of literals. Exercises \(7-11\) are concerned with representations of this kind. Show that the Boolean sum \(y_{1}+y_{2}+\cdots+y_{n},\) where \(y_{i}=\) \(x_{i}\) or \(y_{i}=\overline{x}_{i},\) has the value 0 for exactly one combination of the values of the variables, namely, when \(x_{i}=0\) if \(y_{i}=x_{i}\) and \(x_{i}=1\) if \(y_{i}=\overline{x}_{i} .\) This Boolean sum is called a maxterm.

Construct a circuit that compares the two-bit integers \(\left(x_{1} x_{0}\right)_{2}\) and \(\left(y_{1} y_{0}\right)_{2},\) returning an output of 1 when the first of these numbers is larger and an output of 0 otherwise.

Suppose that \(F\) is a Boolean function represented by a Boolean expression in the variables \(x_{1}, \ldots, x_{n} .\) Show that \(F^{d}\left(x_{1}, \ldots, x_{n}\right)=\overline{F\left(\overline{x}_{1}, \ldots, \overline{x}_{n}\right)}\)

Show that cells in a K-map for Boolean functions in five variables represent minterms that differ in exactly one literal if and only if they are adjacent or are in cells that become adjacent when the top and bottom rows and cells in the first and eighth columns, the first and fourth columns, the second and seventh columns, the third and sixth columns, and the fifth and eighth columns are considered adjacent.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.