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Chapter 2: Problem 35

A circuit has four inputs and two outputs. The inputs \(A_{3: 0}\) represent a number from 0 to 15 . Output \(P\) should be TRUE if the number is prime ( 0 and 1 are not prime, but \(2,3,5\), and so on, are prime). Output \(D\) should be TRUE if the number is divisible by 3 . Give simplified Boolean equations for each output and sketch a circuit.

Short Answer

Expert verified
Outputs P and D have Boolean equations: P = A_0A_1'A_2'A_3' + ... and D = A_0'A_1'A_2'A_3' + ...; sketch uses these equations.

Step by step solution

01

Identifying Prime Numbers

First, determine which numbers between 0 and 15 are prime numbers. These are 2, 3, 5, 7, 11, and 13. This information will help us define the Boolean equation for Output P.
02

Writing Boolean Equation for Output P

For Output P to be TRUE, the input should correspond to a prime number. The truth table output will be TRUE for inputs corresponding to the decimal values of 2, 3, 5, 7, 11, and 13. Convert these decimal values to binary and create the Boolean equation: \[P = A_3'A_2'A_1A_0 + A_3'A_2A_1'A_0 + A_3'A_2A_1A_0' + A_3A_2'A_1A_0' + A_3A_2A_1'A_0 + A_3A_2A_1A_0'\].
03

Simplifying Boolean Equation for Output P

Simplify the equation using Boolean algebra rules. After simplification, the equation could look like:\[P = A_0A_1'A_2'A_3' + A_0'A_1A_2'A_3' + A_0A_1'A_2A_3' + A_0'A_1A_2A_3 + A_0A_1A_2'A_3\].
04

Identifying Numbers Divisible by 3

Next, we identify which numbers between 0 and 15 are divisible by 3. These numbers are 0, 3, 6, 9, 12, and 15.
05

Writing Boolean Equation for Output D

For Output D to be TRUE, the input should correspond to numbers divisible by 3. The truth table output will be TRUE for inputs corresponding to these decimal values. Convert them to binary and write the Boolean equation: \[ D = A_3'A_2'A_1'A_0' + A_3'A_2A_1'A_0 + A_3'A_2A_1A_0' + A_3A_2'A_1A_0' + A_3A_2A_1'A_0' + A_3A_2A_1A_0 \].
06

Simplifying Boolean Equation for Output D

Simplify the equation using Boolean algebra rules to obtain the simplest version, which could be: \[ D = A_0'A_1'A_2'A_3' + A_0A_1'A_2'A_3' + A_0'A_1A_2 + A_0'A_1'A_2 \].
07

Sketching the Circuit

To sketch the circuit, use logic gates to represent the simplified Boolean equations for P and D. Use AND, OR, and NOT gates accordingly to achieve the outputs based on the input connections that satisfy the equations. Draw separate circuits for P and D based on their Boolean equations.

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Key Concepts

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

Prime Numbers
When working with concepts like Boolean Algebra in circuit design, understanding prime numbers is crucial since it affects logical operations and outcomes in your design, particularly for output P in our exercise. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Between 0 and 15, the prime numbers are 2, 3, 5, 7, 11, and 13.
  • Primes like 2 and 5 are easy to spot because they have simple divisibility rules.
  • In electronic circuit design, detecting primes can streamline logic gate configurations which lead to the output being true only for these inputs.
In our exercise, identifying these numbers assists with setting the true condition for our output P, helping inform how Boolean equations are created and used.
Divisibility Rules
Divisibility rules help in determining whether a number can be divided by another without leaving a remainder. This is essential in Boolean logic where certain inputs trigger true conditions. For output D of our digital circuit, we're interested in numbers divisible by 3 between 0 and 15. These numbers are determined as 0, 3, 6, 9, 12, and 15.
  • For a number to be divisible by 3, the sum of its digits must be divisible by 3.
  • In a Boolean expression, these numbers translate into specific binary inputs that simplify circuit design.
Implementing divisibility checks in circuit design uses simple algorithms and enhances how efficiently a particular circuit behaves under certain input conditions.
Logic Gates
Logic gates are the fundamental building blocks of digital circuits. Understanding AND, OR, and NOT gates is crucial for translating Boolean expressions into physical circuits. Each type of gate performs a basic logical function:
  • AND gates output true only when all inputs are true.
  • OR gates output true if at least one input is true.
  • NOT gates invert the input signal.
In our exercise, these gates are used to implement the Boolean expressions for the outputs P and D. This means arranging them so that the outputs signal true based on the prime checks and divisibility rules derived from input data.
Circuit Design
Circuit design involves arranging logic gates to fulfill specific requirements set by Boolean expressions. The challenge is simplifying these expressions to use fewer gates, reducing complexity, and improving efficiency. For the given outputs P and D:
  • Each true condition related to prime numbers and divisibility must connect with respective logic gate configurations.
  • Simplifying Boolean equations minimizes the physical resources needed.
Once gates are connected according to simplified equations, we get a clear layout of how inputs result in desired outputs through electrical paths, ensuring that the digital circuit operates under the specified rules.

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Most popular questions from this chapter

Simplify the following Boolean equations using Boolean theorems. Check for correctness using a truth table or K-map. (a) \(Y=A C+\bar{A} \bar{B} C\) (b) \(Y=\bar{A} \bar{B}+\bar{A} \bar{B} \bar{C}+(\overline{A+\bar{C}})\) (c) \(Y=\bar{A} \bar{B} \bar{C} \bar{D}+A \bar{B} \bar{C}+A \bar{B} C \bar{D}+A B D+\bar{A} \bar{B} C \bar{D}+B \bar{C} D+\bar{A}\)

Alyssa P. Hacker says that any Boolean function can be written in minimal sum- of-products form as the sum of all of the prime implicants of the function. Ben Bitdiddle says that there are some functions whose minimal equation does not involve all of the prime implicants. Explain why Alyssa is right or provide a counterexample demonstrating Ben's point.
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