Discrete Mathematics

1. Boolean Algebra
   1. Boolean Operations
      1. Basic Operations
         1. AND (Conjunction)
            1. Definition: True if both operands are true
            2. Symbol: ∧
            3. Truth Table Analysis
         2. OR (Disjunction)
            1. Definition: True if at least one operand is true
            2. Symbol: ∨
            3. Truth Table Analysis
         3. NOT (Negation)
            1. Definition: Inverts the truth value
            2. Symbol: ¬
            3. Truth Table Analysis
      2. Derived Operations
         1. NAND (Not AND)
            1. Definition: True if not both operands are true
            2. Symbol: ↑
            3. Truth Table Analysis
         2. NOR (Not OR)
            1. Definition: True if neither operand is true
            2. Symbol: ↓
            3. Truth Table Analysis
         3. XOR (Exclusive OR)
            1. Definition: True if exactly one operand is true
            2. Symbol: ⊕
            3. Truth Table Analysis
         4. XNOR (Exclusive NOR)
            1. Definition: True if the operands are equally valued
            2. Symbol: ⊙
            3. Truth Table Analysis
   2. Boolean Functions
      1. Representation of Boolean Functions
         1. Using Truth Tables
            1. Construction of Truth Tables
            2. Interpretation of Truth Tables
         2. Algebraic Normal Form (ANF)
            1. Definition and Formation
            2. Examples of ANF
         3. Canonical Forms
            1. Sum of Products (SOP)
               1. Definition and Examples
               2. Conversion from Truth Table to SOP
            2. Product of Sums (POS)
               1. Definition and Examples
               2. Conversion from Truth Table to POS
      2. Simplification Techniques
         1. Algebraic Simplification
            1. Basic Identities
               1. Idempotent Law
               2. Domination Law
               3. Identity Law
               4. Complement Law
            2. Associative, Commutative, and Distributive Laws
            3. Absorption and De Morgan's Law
         2. Karnaugh Maps
            1. Construction and Use
            2. Minimization of Boolean Functions
            3. Handling Don't Care Conditions
         3. Quine-McCluskey Method
            1. Algorithm and Steps
            2. Comparison with Karnaugh Maps
   3. Applications of Boolean Algebra
      1. Digital Logic Design
         1. Logic Gates
            1. Basic Gates (AND, OR, NOT)
            2. Universal Gates (NAND, NOR)
         2. Combinational Circuits
            1. Adders (Half Adder, Full Adder)
            2. Multiplexers and Demultiplexers
            3. Encoders and Decoders
         3. Sequential Circuits
            1. Flip-Flops (SR, D, JK, T)
            2. Latches
            3. Registers and Counters
      2. Circuit Minimization
         1. Importance in Hardware Design
         2. Techniques for Reducing Gate Count
         3. Trade-offs between Complexity, Speed, and Cost
      3. Programming and Software Development
         1. Use in Conditional Statements and Control Flow
         2. Implementation in Logical Programming and Constraint Solving
      4. Switching Theory
         1. Basis for the Design of Digital Switching Circuits
         2. Analysis of Logic Networks
         3. Design Case Studies: Exemplar Applications in Real-World Systems