Overview
This lecture covers the fundamentals of Boolean logic, including logical operators, truth tables, logic gates, Boolean algebra laws, and De Morgan's theorems.
Core Concepts of Boolean Logic
- Boolean logic is based on binary decisions with two possible truth values: true (1) or false (0).
- A truth table lists all possible combinations of variables and their resulting output.
- Tautology is a logical expression that always results in true; fallacy always results in false.
Logical Operators
- Logical operators combine variables/constants (0 and 1): NOT, OR, AND.
- NOT is a unary operator (complements a single value); symbolized as a bar (e.g., X̅).
- OR is a binary operator (output is 1 if any input is 1); symbolized as plus (+).
- AND is a binary operator (output is 1 only if all inputs are 1); symbolized as dot (·).
Truth Table Examples
- For each operator, truth tables show outputs for all possible input combinations.
- Example expressions can be evaluated stepwise using truth tables to find their outputs.
Logic Gates in Circuits
- Logic gates are electronic circuits performing Boolean operations.
- Basic gates: NOT (inverter), OR, and AND.
- Derived gates: NOR (NOT OR), NAND (NOT AND), XOR (exclusive OR), XNOR (exclusive NOR).
- Symbols, inputs, and truth tables distinguish each gate.
Boolean Algebra Postulates & Laws
- Boolean variables take only 0 or 1.
- Four postulates: binary value, OR, AND, complement.
- Principle of duality: interchange AND/OR and 0/1 to get dual expressions.
Common Boolean Laws
- Idempotent Law: X+X = X; X·X = X.
- Involution Law: (X̅)̅ = X.
- Complementary Law: X+X̅=1; X·X̅=0.
- Commutative, Associative, Distributive, Absorption, and Third Distribution Laws.
- Laws can be proved using truth tables.
De Morgan’s Theorems
- First: (X+Y)̅ = X̅·Y̅ ("break the line, change the sign").
- Second: (XY)̅ = X̅+Y̅.
- These theorems are proved using the previously stated laws.
Drawing Logic Circuits
- Boolean expressions can be converted to logic circuits using basic and derived gates.
- Break down complex expressions into smaller terms and implement sequentially using appropriate gates.
Key Terms & Definitions
- Binary Decision — A decision with two possible outcomes: true/false or 1/0.
- Truth Table — A table showing all input combinations and their corresponding outputs.
- Tautology — An expression always evaluating to true (1).
- Fallacy — An expression always evaluating to false (0).
- Logic Gate — Electronic circuit implementing a Boolean operation.
- Duality — Principle of interchanging operators and identity elements to get a dual expression.
Action Items / Next Steps
- Practice constructing truth tables and evaluating Boolean expressions.
- Draw logic circuits for given Boolean expressions.
- Review Boolean algebra laws and De Morgan’s theorems.
- Prepare for Chapter 4: Introduction to Problem Solving.