Digital Systems and Boolean Algebra Quiz

Digital Systems and Boolean Algebra

• Digital systems are combinations of circuits (electronic devices).
• Circuit behaviors are analyzed using Boolean algebra, a mathematical discipline.
• Boolean algebra is named after George Boole (1815-1864), an English mathematician.
• Boole developed a formal system for working with truth values.
• Boolean logic manipulates true (1) and false (0) values.

Boolean Algebra

• Digital circuits can be represented as black boxes with inputs and outputs.
• Boolean algebra describes the relationship between inputs and outputs using logical operators.
• Basic logical operators include AND, OR, and NOT.

Basic Operators

• NOT Operator: Result is 1 only if the operand is 0. Also known as inversion or negation. Notation: A'
• AND Operator: Result is 1 only if both operands are 1. Also known as logical product. Notation: A • B
• OR Operator: Result is 1 if either operand is 1. Also known as logical sum. Notation: A + B

Useful Laws

• Identity and Annulment Laws: X + 0 = X, X + 1 = 1, X • 1 = X, X • 0 = 0
• Idempotent Law: X + X = X, X • X = X
• Involution Law: (X)' = X
• Complementarity Law: X + X' = 1, X • X' = 0

Commutative, Associative, and Distributive Laws

• Commutative Law: X + Y = Y + X, X • Y = Y • X
• Associative Law: (X + Y) + Z = X + (Y + Z), (X • Y) • Z = X • (Y • Z)
• Distributive Law: X • (Y + Z) = (X • Y) + (X • Z), X + (Y • Z) = (X + Y) • (X + Z)

Boolean Algebra: Theorems

• Expansion Theorem: X • Y + X • Y' = X
• Absorption Theorem: X + X • Y = X

Boolean Algebra: Proving Theorems

• Examples of how some theorems are proven using prior laws.

DeMorgan's Law

• NOR is equivalent to AND with complemented inputs. (X + Y)' = X' • Y'
• NAND is equivalent to OR with complemented inputs. (X • Y)' = X' + Y'

Boolean Algebra: Example of Use

• Simplifying Boolean functions using algebraic laws and theorems.

Abstract vs. Implementation

• Computers use two states: true/false, on/off, 1/0.
• Boolean algebra is used to understand and design electronic circuits.
• Logic circuits use voltages to represent logical values.

Transistors

• Transistors are used to implement binary logic in hardware.
• Computers use many transistors.
• MOS transistors are examples.

MOS Transistor

• MOS transistors have three terminals: Gate, Source, and Drain.
• Gate controls current flow.
• Two types of MOS transistors: N-type and P-type

Different Types of MOS Transistors

• N-type gates are normally open
• P-type gates are normally closed
• Voltages at the gate can control the flow of current

How Does a Transistor Work

• Transistors can be used like switches.
• The presence or absence of current represents TRUE or FALSE.

Logic Gates

• Basic logical units made from MOS transistors.
• Implement logical operations. (NOT, AND, OR, NAND, NOR, XOR)
• Used to build more complex circuits

CMOS NOT Gate (Inverter)

• Shows how complementary MOS transistors are used.

Logic Gates : Symbols, Function, and Truth Tables

• Shows the different logic gates, their symbols, how they work, and what truth values they deliver. Various gates include AND, OR, NAND, NOR, XOR, NOT, etc.
• Includes the symbols and logic functions alongside their respective truth tables.

Logic Circuit (Boolean function)

• Circuits that implement a Boolean function.

Types of Circuit

• Combinational: Output depends only on present input values.
• Sequential: Output depends on present and past input values.

Boolean function vs. logic block

• Boolean function describes the relationship between inputs and outputs
• Logic block (gates) implement those connections.

Boolean function as an Algebraic Expression

• Boolean output functions can be represented as algebraic expressions.
• The symbols used within the algebraic expression are called literals.
• The use of literal values may result in more than one algebraic expressions for a given Boolean function.

Boolean function as a Truth Table

• Truth table displays all possible input combinations and resulting output values for a Boolean function.
• The output value can be a 1 or 0 in a given row.

Canonical Forms of Boolean function

• Standard forms for algebraic expressions, like Sum of Products (SOP) and Product of Sums (POS).
• Useful for simplifying and representing Boolean functions with many inputs.

Sum of Products Form (SOP)

• Method for expressing Boolean functions using AND and OR operations on the inputs.
• Includes a method for creating the SOP representation of functions using a truth table

Shorthand notation for canonical SOP

• Provides a concise way to represent Boolean expressions in SOP form.
• Uses decimal numbers in conjunction with the letter "m" and the input variable letters to describe the inputs and result
• Allows for the quick comparison of the output function outcome

Canonical SOP vs Minimal SOP

• Canonical SOP : standard form using all possible combinations of input values
• Minimal SOP: simplified form using the fewest possible terms

Canonical SOP Forms

• SOP method to express Boolean functions in two levels using AND and OR logical gates.

Product Of Sums Form (POS)

• Method of expressing Boolean functions using OR and AND operations for the inputs. Includes a method for creating the POS representation of functions using a truth table

Shorthand Notation of canonical POS

• Provides a concise way to represent Boolean expressions in POS form.
• Uses decimal numbers in conjunction with the letter "M" and the input variable letters to describe the inputs and result
• Allows for the quick comparison of the output function outcome

Useful Conversions SOP/POS

• Methods to convert between SOP and POS forms of Booelan expressions.

Logic circuit Designing: Steps to follow

• Steps in creating a design from a logic specification statement through to an implementation.
• Includes a description of how to solve a problem in logic circuit design using a truth table and building out the logic equations through simplification (SOP format) to a final circuit diagram representation

Logic Simplification: Karnaugh Maps (K-Maps)

• Graphical method for simplifying Boolean expressions.

Quick Recap on Logic Simplification

• Original Boolean functions are sometimes too complex
• Boolean algebra and K-maps can simplify operations and reduce the amount of circuitry needed

Logic Simplification

• Techniques used to simplify Boolean expressions, such as the Uniting Theorem.

Karnaugh Map (K-map) method

• Method of visualising and simplifying Boolean functions, based on the Uniting Theorem
• Provides a different way of representing a truth table
• Uses gray code

K-map Simplification for Three Variables (1 and 2)

• Method for simplifying Boolean functions with three variables using K-maps.

K-maps with “Don't Care”

• Don't Care input values are useful to create simpler circuit implementations.
• Practical circuits may ignore inputs that are unlikely to ever occur

Example 1: Seven-segment Decoder

• Illustrates the use of don't cares in simplifying circuit design (specific to segment implementations)

Example 2: BCD Increment Function

• Example of don't care usage in BCD circuits (specific example in circuit design).

K-map for BCD Increment Function

• Applying the K-map method to simplify the design of a BCD increment circuit (showing examples to simplify circuits).

Test your knowledge on digital systems and the principles of Boolean algebra. This quiz covers basic operators like AND, OR, and NOT, as well as their applications in circuit behaviors. Explore how these concepts are foundational to understanding electronic devices.