Book cover for Discrete Mathematics with Applications

Discrete Mathematics with Applications

Thomas Koshy

ISBN #124211801

1st Edition

3,631 Questions

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38,965 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter highlights the profound impact of Boolean algebra on the design and simplification of digital circuits. Starting from its theoretical roots in symbolic logic through George Boole’s foundational work, Boolean algebra now underpins modern electronic circuit design. Through practical examples such as switch toggles, adders, and display drivers, the chapter illustrates how Boolean logic enables the efficient realization of complex circuit functionalities while emphasizing simplification and optimization.

Learning Objectives

1

Explain the historical development of Boolean algebra and its significance in modern digital circuit design.

2

Analyze and design combinational circuits using Boolean algebra, including toggle circuits, adders, and display drivers.

3

Apply Boolean algebra principles to simplify complex electronic circuits without sacrificing functionality.

4

Demonstrate the practical applications of Boolean algebra in real-world scenarios such as lighting control and digital displays.

Key Concepts

CONCEPT

DEFINITION

Boolean Algebra

A mathematical system that uses binary variables and logical operations such as AND, OR, and NOT to represent and simplify logical expressions and circuits.

Combinational Circuits

Electronic circuits whose output is determined solely by the current input values, without any memory of past inputs.

XOR (Exclusive OR)

A logical operation that outputs true only when the number of true inputs is odd; commonly used to design toggle circuits.

Digital Logic

The area of study in electronics and computer science that focuses on representing logical functions using binary numbers and logic gates.

Seven-Segment Display

A type of electronic display that uses up to seven individual segments to represent decimal digits, commonly driven by binary-coded decimal signals.

Example Problems

Example 1

Using Example $12.2,$ evaluate each. $$6+10$$

Example 2

Using Example $12.2,$ evaluate each. $$6.10$$

Example 3

Using Example $12.2,$ evaluate each. $$2+(3 + 5)$$

Example 4

Using Example $12.2,$ evaluate each. $$(2+3) + 5$$

Example 5

Using Example $12.2,$ evaluate each. $$3.(5.6)$$
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Step-by-Step Explanations

QUESTION

How can Boolean algebra be used to design a circuit that toggles a light on or off using three switches?

STEP-BY-STEP ANSWER:

Step 1: Represent each switch with a binary variable (e.g., A, B, C), where each variable can be 0 (off) or 1 (on).
Step 2: Recognize that the light toggles state when an odd number of switches change state. This behavior is represented by the XOR operation.
Step 3: Formulate the Boolean expression for the light output as L = A ⊕ B ⊕ C.
Step 4: Verify the function by checking that flipping any one switch (altering an odd number of inputs) changes the state of L.
Final Answer: The Boolean expression for the three-switch light circuit is L = A ⊕ B ⊕ C.

Three-Switch Light Circuit

QUESTION

How can Boolean algebra be used to design a circuit that computes the sum of two 3-bit numbers?

STEP-BY-STEP ANSWER:

Step 1: Represent each bit of the two 3-bit numbers as A2 A1 A0 and B2 B1 B0, respectively.
Step 2: Use full adders for each bit position, where each full adder computes a sum bit and a carry-out based on the two input bits and a carry-in.
Step 3: Express the sum bit with the equation: Sum = A ⊕ B ⊕ Carry_in.
Step 4: Express the carry-out with the Boolean formula: Carry_out = (A AND B) OR (Carry_in AND (A ⊕ B)).
Step 5: Cascade three full adders, where the carry-out from one becomes the carry-in of the next.
Final Answer: The circuit using three full adders, governed by the above Boolean equations, will compute the 3-bit sum.

3-Bit Adder Circuit

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Common Mistakes

  • Confusing the XOR operation with a standard OR, which can result in an incorrect representation of toggle functions.
  • Overlooking the importance of carry propagation in designing multi-bit adder circuits.
  • Failing to use Boolean identities for circuit simplification, thereby not optimizing the design.
  • Misinterpreting the role of binary-coded decimal inputs in driving seven-segment displays.