JKoch notes

lecture_EE1D1_lec2_intro_2025-09-02

8 min read

πŸ“Œ Overview

This lecture introduces the foundational concepts of Boolean algebra, essential for understanding digital logic and computer science. We’ll cover basic Boolean operations, their truth tables, and corresponding logic gates, ensuring a strong base for future exercises.

🎯Learning Objectives

  • Explain the importance of digital systems βœ… 2025-09-28
  • Describe the benefits of digital systems over analog systems βœ… 2025-09-28
  • Get familiar with binary systems and basic Boolean algebra βœ… 2025-09-28
  • Convert numbers between different base systems (e.g., from decimal to binary and vice versa) βœ… 2025-09-28
  • Compute with negative and positive numbers βœ… 2025-09-28

πŸ’‘Key Concepts & Definitions

Boolean Algebra

A branch of algebra where variables can only have two truth values: true (1) or false (0). It’s fundamental to digital circuits and logic design.

Boolean Variable

A variable that can hold only one of two possible values, typically denoted as 0 (False) or 1 (True).

Truth Table

A mathematical table used in logic to compute the functional values of logical expressions on all possible combinations of their arguments.

Logic Gate

An elementary building block of a digital circuit that implements a Boolean function.

βž— Formulas

Basic Boolean Identities

  • AND Gate (Logical Conjunction): A AND B = A β‹… B
  • OR Gate (Logical Disjunction): A OR B = A + B
  • NOT Gate (Logical Negation): NOT A = A' or Δ€

Commutative Laws

  • A β‹… B = B β‹… A
  • A + B = B + A

Associative Laws

  • (A β‹… B) β‹… C = A β‹… (B β‹… C)
  • (A + B) + C = A + (B + C)

Distributive Laws

  • A β‹… (B + C) = (A β‹… B) + (A β‹… C)
  • A + (B β‹… C) = (A + B) β‹… (A + C)

De Morgan’s Laws

  • !(A β‹… B) = !A + !B or (A β‹… B)' = A' + B'
  • !(A + B) = !A β‹… !B or (A + B)' = A' β‹… B'

✍️ Notes

Boolean Algebra is a branch of algebra in which the values of the variables are the truth values true and false, usually denoted with 1 and 0 respectively. It was introduced by George Boole in his 1854 book β€œAn Investigation of the Laws of Thought”.

Key Principles:

  • Binary Values: Variables can only take two states (0 or 1).
  • Logical Operations: Operations like AND, OR, NOT are defined.
  • Axioms and Theorems: Follows a set of rules and identities for manipulation.

Importance:

  • Digital Circuits: Fundamental for designing and analyzing digital electronic circuits (e.g., in computers, smartphones).
  • Set Theory: Closely related to set operations (intersection, union, complement).
  • Computer Science: Used in programming, database queries, and artificial intelligence.

Solving/simplifying

Logic Gates

Logic gates are the fundamental building blocks of any digital system. They are electronic circuits that perform basic Boolean functions. Most logic gates have two inputs and one output. The state of the output is determined by the state of the inputs based on the Boolean operation they represent.

Key Characteristics:

  • Digital Operations: Process binary signals (0s and 1s).
  • Building Blocks: Combine to form complex digital circuits, microprocessors, and memory.
  • Implemented using: Diodes, transistors, and resistors.

Basic Logic Gates:

1. AND Gate

  • Function: Output is HIGH (1) only if all inputs are HIGH.
  • Symbol:
\begin{tikzpicture}
\draw (0,0) node[and gate, draw, logic gate inputs=2] {};
\node at (0.9,0) {A β‹… B};
\end{tikzpicture}

2. OR Gate

  • Function: Output is HIGH (1) if any input is HIGH.
  • Symbol:
\begin{tikzpicture}
\draw (0,0) node[or gate, draw, logic gate inputs=2] {};
\node at (0.9,0) {A + B};
\end{tikzpicture}

3. NOT Gate (Inverter)

  • Function: Output is the inverse of the input.
  • Symbol:
\begin{tikzpicture}
\draw (0,0) node[not gate, draw] {};
\node at (0.6,0) {A'};
\end{tikzpicture}

Universal Gates:

4. NAND Gate

  • Function: The NOT of AND. Output is LOW (0) only if all inputs are HIGH.
  • Symbol:
\begin{tikzpicture}
\draw (0,0) node[nand gate, draw, logic gate inputs=2] {};
\node at (0.9,0) {(A β‹… B)'};
\end{tikzpicture}

5. NOR Gate

  • Function: The NOT of OR. Output is HIGH (1) only if all inputs are LOW.
  • Symbol:
\begin{tikzpicture}
\draw (0,0) node[nor gate, draw, logic gate inputs=2] {};
\node at (0.9,0) {(A + B)'};
\end{tikzpicture}

Exclusive Gates:

6. XOR Gate (Exclusive OR)

  • Function: Output is HIGH (1) if inputs are different.
  • Symbol:
\begin{tikzpicture}
\draw (0,0) node[xor gate, draw, logic gate inputs=2] {};
\node at (0.9,0) {A βŠ• B};
\end{tikzpicture}

πŸ“ Summary

Boolean operations are fundamental logical operations on binary values (true/false, 1/0). They form the basis of digital circuits and computer logic.

πŸ’‘ Explanation

Boolean operations manipulate binary inputs to produce a single binary output. They are crucial for designing and understanding digital systems.

Key Operations:

  1. AND (∧):
  • Output is true (1) only if ALL inputs are true (1).
  • A AND B | A | B | A AND B | |---|---|---------| | 0 | 0 | 0 | | 0 | 1 | 0 | | 1 | 0 | 0 | | 1 | 1 | 1 |
  1. OR (∨):
  • Output is true (1) if AT LEAST ONE input is true (1).
  • A OR B | A | B | A OR B | |---|---|--------| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 1 |
  1. NOT (Β¬):
  • Output is the INVERSE of the input.
  • NOT A | A | NOT A | |---|-------| | 0 | 1 | | 1 | 0 |

Example: Logic Gate Application

Consider a simple circuit: (A AND B) OR (NOT C)

ABCA AND BNOT C(A AND B) OR (NOT C)
000011
001000
010011
011000
100011
101000
110111
111101

πŸ–ΌοΈ Diagrams & Visuals

A simple logic circuit demonstrating AND, OR, and NOT gates:

graph TD
   A[Input A] --> G1
   B[Input B] --> G1
   G1(AND Gate) --> G3
   C[Input C] --> G2
   G2(NOT Gate) --> G3
   G3(OR Gate) --> Output

πŸ“š Related Resources

πŸ”— Resources

  • Presentation:

❓ Post lecture

πŸ“– Homework

Truth Tables

A truth table is a mathematical table used in logic (especially Boolean algebra, Boolean functions, and propositional calculus) to compute the functional values of logical expressions on all possible combinations of their arguments.

Purpose:

  • Define Logic Gates: Clearly shows the output for every possible input combination of a logic gate (e.g., AND, OR, NOT).
  • Evaluate Boolean Expressions: Helps in determining the truth value of complex Boolean expressions.
  • Verify Equivalence: Used to check if two different Boolean expressions are logically equivalent.

How to Construct:

  1. Identify Inputs: Determine all independent Boolean variables (e.g., A, B, C).
  2. Determine Number of Rows: For n input variables, there will be 2^n rows in the table (representing all unique combinations).
  3. List Input Combinations: Systematically list all 2^n input combinations. A common method is to count in binary.
  4. Evaluate Sub-expressions: If the expression is complex, evaluate intermediate parts step-by-step in separate columns.
  5. Determine Final Output: Compute the final output of the entire expression for each input combination.

Example: A AND (NOT B)

Here, n=2 inputs (A, B), so 2^2 = 4 rows.

ABNOT B (B’)A β‹… B’
0010
0100
1011
1100

πŸ“ Summary: Basics of Boolean Operation

This lecture provided a fundamental introduction to Boolean algebra, which is critical for understanding digital systems.

  1. Introduction to Boolean Algebra:

    • Purpose: Deals with variables that can only be True (1) or False (0).
    • Application: Forms the basis of digital circuit design, computer logic, and programming.

  2. Basic Boolean Operations:

    • AND (β‹…): Output is True only if all inputs are True.
      • 0 β‹… 0 = 0, 0 β‹… 1 = 0, 1 β‹… 0 = 0, 1 β‹… 1 = 1
    • OR (+): Output is True if any input is True.
      • 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 1
    • NOT (’): Inverts the input; if input is True, output is False, and vice versa.
      • 0' = 1, 1' = 0

  3. Derived Boolean Operations:

    • XOR (βŠ•) - Exclusive OR: Output is True if inputs are different.
      • 0 βŠ• 0 = 0, 0 βŠ• 1 = 1, 1 βŠ• 0 = 1, 1 βŠ• 1 = 0
    • NAND: NOT of AND. Output is False only if all inputs are True.
    • NOR: NOT of OR. Output is True only if all inputs are False.

  4. Truth Tables:

    • A systematic way to list all possible input combinations and their corresponding outputs for a Boolean expression or logic gate.
    • Example (AND Gate):
      ABA β‹… B
      000
      010
      100
      111

  5. Introduction to Logic Gates:

    • Physical implementations of Boolean operations using electronic circuits.
    • Common Gates: AND, OR, NOT, XOR, NAND, NOR.
    • These gates are the fundamental building blocks of all digital devices.

<smtcmp_block language=β€œmermaid”> graph TD subgraph Basic Logic Gates A[Input A] β‡’ AND1 B[Input B] β‡’ AND1 AND1(AND) β‡’ C[Output C]

    D[Input D] --> OR1
    E[Input E] --> OR1
    OR1(OR) --> F[Output F]

    G[Input G] --> NOT1
    NOT1(NOT) --> H[Output H]
end

Graph View

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