Boolean Algebra Chapter 2

Capítulo 2. Álgebra de Boole

Introducción

• Boolean algebra is a vital tool for designing and analyzing digital systems.
• It facilitates the representation and manipulation of logical functions using algebraic equations.

Variables booleanas

• A Boolean variable can only assume one of two values.
• These values are True (1) and False (0).

Operadores lógicos básicos

• The basic logical operators are AND, OR, and NOT.
• AND is represented by a dot (·) or the absence of an operator.
• OR is represented by the plus sign (+).
• NOT is represented by an overbar (￣) or an apostrophe (').

Tablas de verdad

• Truth tables define the behavior of logical operators for all possible input value combinations.

Tabla de verdad del operador AND

• Shows the output results for all combinations of A and B inputs using the AND operator.
• When A and B are 0, the result is 0.
• When A is 0 and B is 1, the result is 0.
• When A is 1 and B is 0, the result is 0.
• When A and B are 1, the result is 1.

Tabla de verdad del operador OR

• Shows the output results for all combinations of A and B inputs using the OR operator.
• When A and B are 0, the result is 0.
• When A is 0 and B is 1, the result is 1.
• When A is 1 and B is 0, the result is 1.
• When A and B are 1, the result is 1.

Tabla de verdad del operador NOT

• Shows the output when using the NOT operator on A.
• When A is 0, the result is 1.
• When A is 1, the result is 0.

Postulados del álgebra de Boole

• Boolean algebra postulates are a set of axioms defining the basic properties of logical operations.
• P1: Boolean algebra is a closed algebraic system.
• P2: Existence of neutral elements: A · 1 = A for AND, and A + 0 = A for OR.
• P3: Commutativity: A · B = B · A and A + B = B + A.
• P4: Distributivity: A · (B + C) = A · B + A · C and A + (B · C) = (A + B) · (A + C).
• P5: Existence of complements: A · Ā = 0 and A + Ā = 1.

Teoremas del álgebra de Boole

• Theorems are propositions that can be derived from the postulates.

Teoremas fundamentales

• T1: Idempotence: A · A = A and A + A = A.
• T2: Absorption: A · (A + B) = A and A + (A · B) = A.
• T3: Simplification: A · (Ā + B) = A · B and A + (Ā · B) = A + B.
• T4: DeMorgan's Laws: overline{A \cdot B} = overline{A} + overline{B} and overline{A + B} = overline{A} \cdot overline{B}.

Teoremas adicionales

• T5: Associativity: (A · B) · C = A · (B · C) and (A + B) + C = A + (B + C).
• T6: Involution: overline{overline{A}} = A.
• T7: Consensus: A · B + Ā · C + B · C = A · B + Ā · C and (A + B) · (Ā + C) · (B + C) = (A + B) · (Ā + C).

Funciones booleanas

• It's an algebraic expression that describes the relationship between input Boolean variables and a single Boolean output variable.

Representación de funciones booleanas

• Boolean functions can be represented algebraically, using truth tables, or with logic diagrams.
• Algebraic Expression: An equation using basic logical operators.
• Truth Table: A table showing function values for all possible input value combinations.
• Logic Diagram: A circuit implementing the function using logic gates.

Formas canónicas

• Canonical forms are normalized representations.
• Sum of Products (SOP): The function as a sum of product terms, each containing all input variables, either in direct or complemented form.
• Product of Sums (POS): The function as a product of sum terms, each containing all input variables, either in direct or complemented form.

Simplificación de funciones booleanas

• The process of finding the simplest algebraic expression that represents the same function.

Métodos de simplificación

• Different methods for simplifying Boolean functions include:
• Boolean Algebra: Applying postulates and theorems to reduce algebraic expression complexity.
• Karnaugh Maps: Diagrams that visually identify redundant terms in the function.
• Quine-McCluskey Method: A systematic algorithm to find the function's minimum expression.

Importancia de la simplificación

• Boolean simplification is important because it reduces logic gates, decreases circuit cost and size, and improves performance and efficiency.

Atomic Habits

James Clear

Introduction

• This book is an applied guide to building good habits and breaking bad ones, based on a proven system.

My Story

• The author recounts a high school accident involving a baseball bat that led to severe injuries and a medically induced coma.
• The recovery process involved relearning basic tasks and gradually improving physical and academic performance.
• Consistent effort and small improvements led to significant achievements, including making the baseball team and excelling in college.

How?

• Tiny, consistent improvements lead to astounding differences over time.
• Improving by 1 percent each day for a year results in being thirty-seven times better.
• Conversely, declining by 1 percent each day leads to near-zero results.
• Habits compound over time, either positively or negatively.

Why Tiny Changes Make a Big Difference

Forget About Setting Goals, Focus on Your System Instead.

• Goals focus on the desired outcome, while systems focus on the processes to achieve the outcome.
• Goals can be problematic because winners and losers may share the same goals, achieving a goal is temporary, goals restrict happiness, and goals conflict with long-term progress.
• Systems are about continuous refinement and improvement, emphasizing commitment to the process over single accomplishments.

Identity-Based Habits

The most effective way to change your habits is to focus not on what you want to achieve, but on who you wish to become.

• Identity-based habits focus on changing your identity rather than just achieving outcomes.

Two Steps to change your identity:

• First, decide the type of person you want to be.
• Second, prove it to yourself with small wins reinforcing the desired identity.
• Example: Adopting the identity of a reader by starting with small actions like reading one page a day.

The Four Laws of Behavior Change

• Cue, craving, response, and reward form the backbone of every habit, processed sequentially by the brain.

The 1st Law: Make It Obvious

• Increase awareness of habits by writing them out in a Habit Scorecard, listing daily routines.

Implementation Intention

• Using the formula "I will [BEHAVIOR] at [TIME] in [LOCATION]" to create clear plans for habit execution.

Habit Stacking

• Using the formula "After [CURRENT HABIT], I will [NEW HABIT]" to link new habits to existing routines.

The 2nd Law: Make It Attractive

• Temptation bundling involves pairing a needed habit with a desired one.

Temptation Bundling

• Example: Combining Netflix watching with exercise.

The 3rd Law: Make It Easy

• Differentiating between motion (planning) and action (actually doing).

The Two-Minute Rule

• Scaling down habits to take two minutes or less to make them easy to start.
• Examples: Reading one page instead of reading before bed, taking out a yoga mat instead of doing thirty minutes of yoga.

The 4th Law: Make It Satisfying

• Emphasizes the importance of immediate rewards for repeated behavior and immediate punishment for avoided behavior.

Habit Tracker

• Using habit trackers to measure consistency in performing a habit, often via crossing off days on a calendar.

Never Miss Twice

• Getting back on track as quickly as possible after missing a habit.

Advanced Tactics

• Genes don't determine destiny but rather areas of opportunity, and people perform differently when being observed.

Goldilocks Zone

• Humans experience peak motivation when working on tasks that are right on the edge of their current abilities.