Discrete Mathematics

Discrete Mathematics is a branch of mathematics that deals with countable, distinct, and separate structures rather than continuous ones. It encompasses a variety of topics including combinatorics, graph theory, set theory, and algorithms, focusing on concepts that are foundational for computer science and theoretical mathematics. Discrete Mathematics provides the essential tools for analyzing finite systems and is pivotal in fields such as cryptography, network design, and optimization, where discrete structures are prevalent.

1. Foundations of Discrete Mathematics
   1. Logic
      1. Propositional Logic
         1. Logical Connectives
            1. Conjunction (AND)
            2. Disjunction (OR)
            3. Negation (NOT)
            4. Implication (IF-THEN)
            5. Biconditional (IF AND ONLY IF)
         2. Truth Tables
            1. Definition and Construction
            2. Application in Determining Logical Validity
         3. Logical Equivalence
            1. Identity Laws
            2. Domination Laws
            3. Idempotent Laws
            4. Double Negation Law
            5. Commutative Law
            6. Associative Law
            7. Distributive Law
            8. De Morgan’s Laws
            9. Absorption Law
         4. Tautologies and Contradictions
            1. Definition and Examples
            2. Distinction Between Tautology, Contradiction, and Contingency
      2. Predicate Logic
         1. Predicates and Quantifiers
            1. Definition of Predicates
            2. Scope of Quantifiers
         2. Universal and Existential Quantifiers
            1. Universal Quantifier (∀)
            2. Existential Quantifier (∃)
            3. Quantifier Negation and Equivalences
         3. Logical Inference
            1. Rules of Inference
            2. Validity of Arguments
            3. Proof Techniques (Direct, Indirect)
   2. Set Theory
      1. Basic Set Operations
         1. Union (A ∪ B)
            1. Definition and Properties
         2. Intersection (A ∩ B)
            1. Definition and Properties
         3. Difference (A - B)
            1. Definition and Properties
         4. Complement (A')
            1. Definition and Properties
      2. Venn Diagrams
         1. Representation of Sets and Their Relationships
         2. Application in Solving Logical and Set Theoretical Problems
      3. Cartesian Products
         1. Definition and Examples
         2. Properties and Application
      4. Power Sets
         1. Definition and Calculation
         2. Properties
      5. Cardinality and Countability
         1. Finite and Infinite Sets
         2. Countable and Uncountable Sets
         3. Concept of Cardinality
   3. Functions and Relations
      1. Functions
         1. One-to-One (Injective) Functions
            1. Definition and Examples
            2. Horizontal Line Test
         2. Onto (Surjective) Functions
            1. Definition and Examples
            2. Restricting the Codomain
         3. Bijective Functions
            1. Definition and Examples
            2. Inverse Existence
         4. Inverse Functions
            1. Definition and Examples
            2. Properties of Inverses
         5. Composition of Functions
            1. Definition and Examples
            2. Associative Property
      2. Relations
         1. Properties of Relations
            1. Reflexive
               1. Reflexivity in Context of Relations
            2. Symmetric
               1. Symmetry Condition in Relations
            3. Transitive
               1. Transitivity and Its Implications
         2. Equivalence Relations
            1. Definition and Examples
            2. Equivalence Classes
            3. Properties and Applications
         3. Partial Orderings
            1. Definition and Examples
            2. Posets (Partially Ordered Sets)
            3. Hasse Diagrams