/*! 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 46 Construct a combinatorial circui... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 46

Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output \((p \wedge \neg r) \vee(\neg q \wedge r)\) from input bits \(p, q,\) and \(r .\)

Short Answer

Expert verified
Use NOT gates for \(eg r\) and \(eg q\), AND gates for \((p \wedge eg r)\) and \((eg q \wedge r)\), and an OR gate to combine their outputs.

Step by step solution

01

- Identify the Required Logic Gates

Determine the logic gates required for the given boolean expression \((p \wedge eg r) \vee (eg q \wedge r)\). The expression uses AND (\(\wedge\)), OR (\(\vee\)), and NOT (\(eg\)) operations.
02

- Break Down the Expression

Decompose the expression into smaller parts. There are two main components: - \((p \wedge eg r)\)- \((eg q \wedge r)\)
03

- Construct the NOT Gates

Implement the NOT gates for \(r\) and \(q\). \(eg r\) and \(eg q\) will require two NOT gates, with the inputs being \(r\) and \(q\) respectively.
04

- Construct the AND Gates

Implement the AND gates for each component:- The first AND gate has inputs \(p\) and \(eg r\). - The second AND gate has inputs \(eg q\) and \(r\).
05

- Construct the OR Gate

Implement the OR gate that combines the outputs of the two AND gates. The inputs to the OR gate are the outputs of \((p \wedge eg r)\) and \((eg q \wedge r)\).
06

- Combine All Components

Connect the output of the NOT gates to the respective AND gates, and finally, the outputs of the AND gates to the OR gate to get the final result.

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.

Logic Gates
Logic gates are the building blocks of digital circuits. They perform basic logical functions and are used in the construction of combinatorial circuits. In our exercise, we use three types of logic gates:
  • AND gate: Produces an output of 1 only if both inputs are 1. Represented as \( \wedge \) in Boolean expressions.
  • OR gate: Produces an output of 1 if at least one input is 1. Represented as \( \vee \) in Boolean expressions.
  • NOT gate (Inverter): Produces the opposite value of the input. If the input is 1, the output is 0, and vice versa. Represented as \( \eg \) in Boolean expressions.
In our problem, we need:
  • Three NOT gates to invert the variables \(p\) and \(q\).
  • Two AND gates to combine the inverted inputs with other variables.
  • One OR gate to produce the final output from the results of the AND gates.
Boolean Expressions
Boolean expressions are algebraic expressions used to represent logical relationships in digital circuits. They involve variables and logical operations such as AND, OR, and NOT.
Let's break down the given Boolean expression: \( (p \wedge \eg r) \vee (\eg q \wedge r) \):
  • \( p \wedge \eg r \): This part of the expression means that the output will be 1 if \(p\) is 1 and \(r\) is 0.
  • \( \eg q \wedge r \): This part means that the output will be 1 if \(q\) is 0 and \(r\) is 1.

We then use the OR operation to combine these two parts, meaning the output is 1 if either condition is met.
Circuit Design
Designing a combinatorial circuit involves arranging logic gates to meet the specifications of a given Boolean expression. Here鈥檚 how you step-by-step design the circuit for \( (p \wedge \eg r) \vee (\eg q \wedge r) \):
  • Step 1: Identify the required logic gates - we need AND, OR, and NOT gates.
  • Step 2: Decompose the expression into smaller components. This helps in understanding which gates to connect first.
  • Step 3: Implement the NOT gates for \( r \) and \( q \). This gives us \( \eg r \) and \( \eg q \).
  • Step 4: Implement the AND gates. One AND gate combines \( p \) and \( \eg r \). The other combines \( \eg q \) and \( r \).
  • Step 5: Implement the OR gate which takes the outputs of both AND gates as its input.
  • Step 6: Connect all components accordingly to get the final output.
By following these steps carefully, you can design a circuit that accurately represents any Boolean expression.

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

Let \(Q(x)\) be the statement " \(x+1>2 x\) . If the domain consists of all integers, what are these truth values? $$ \begin{array}{llll}{\text { a) }} & {Q(0)} & {\text { b) } Q(-1)} & {\text { c) }} \quad {Q(1)} \\ {\text { d) }} & {\exists x Q(x)} & {\text { e) } \quad \forall x Q(x)} & {\text { f) } \quad \exists x \neg Q(x)}\end{array} $$ g) \(\quad \forall x \neg Q(x)\)

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) Something is not in the correct place. b) All tools are in the correct place and are in excellent condition. c) Everything is in the correct place and in excellent condition. d) Nothing is in the correct place and is in excellent condition. e) One of your tools is not in the correct place, but it is in excellent condition.

Prove or disprove that you can use dominoes to tile a \(5 \times 5\) checkerboard with three corners removed.

For each of these arguments determine whether the argument is correct or incorrect and explain why. a) All students in this class understand logic. Xavier is a student in this class. Therefore, Xavier understands logic. b) Every computer science major takes discrete math- ematics. Natasha is taking discrete mathematics. Therefore, Natasha is a computer science major. c) All parrots like fruit. My pet bird is not a parrot. Therefore, my pet bird does not like fruit. d) Everyone who eats granola every day is healthy. Linda is not healthy. Therefore, Linda does not eagranola every day.

Suppose that \(a\) and \(b\) are odd integers with \(a \neq b .\) Show there is a unique integer \(c\) such that \(|a-c|=|b-c|\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Discrete 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.