Propositional Logic and Set Theory Quiz

Logical Operators:
- AND (∧): True if both operands are true.
- OR (∨): True if at least one operand is true.
- NOT (¬): Inverts the truth value of the operand.
- IMPLICATION (→): True unless a true operand implies a false operand.
- BICONDITIONAL (↔): True if both operands are either true or false.
Truth Tables:
- A systematic way to determine the truth value of logical expressions based on the truth values of their components.
Tautology: A statement that is always true, regardless of the truth values of its variables.
Contradiction: A statement that is always false.

Basic Set Operations:
- Union ( ∪ ): Combines elements from both sets; A ∪ B = {x | x ∈ A or x ∈ B}.
- Intersection ( ∩ ): Elements common to both sets; A ∩ B = {x | x ∈ A and x ∈ B}.
- Difference ( \ ): Elements in one set that are not in the other; A \ B = {x | x ∈ A and x ∉ B}.
- Complement: Elements not in the set; if U is the universal set, A' = U \ A.
Power Set: The set of all subsets of a set A, including the empty set and A itself; denoted as P(A).
Cardinality: The number of elements in a set. Finite sets have a finite cardinality, while infinite sets have infinite cardinality.

Graphs:
- Definition: A graph G consists of a set of vertices (V) and a set of edges (E) connecting pairs of vertices.
- Types:
  - Undirected Graph: Edges have no direction.
  - Directed Graph (Digraph): Edges have a direction from one vertex to another.
Basic Properties:
- Degree: The number of edges incident to a vertex.
- Path: A sequence of edges that connect a sequence of vertices.
- Cycle: A path that starts and ends at the same vertex without repeating any edges.
Combinatorial Aspects:
- Counting Paths: Techniques such as the multiplication principle, where the number of ways to achieve a sequence of choices is the product of the number of choices at each step.
- Subgraphs: A graph formed from a subset of vertices and edges of another graph.
- Graph Coloring: Assigning colors to vertices such that no two adjacent vertices have the same color; used in scheduling and mapping problems.
Applications: Network design, resource allocation, and social network analysis.

Logical Operators define how truth values are combined:
- AND (∧): True only if both operands are true.
- OR (∨): True if at least one operand is true.
- NOT (¬): Negates the truth value of the operand.
- IMPLICATION (→): Only false if a true operand implies a false operand.
- BICONDITIONAL (↔): True if both operands share the same truth value.
Truth Tables are systematic tools to evaluate the truth values of logical expressions based on their components.
Tautology refers to statements that are always true, regardless of variable truth values.
Contradiction denotes statements that are invariably false.

Basic Set Operations are fundamental methods for manipulating sets:
- Union ( ∪ ): Combines elements from two sets, represented as A ∪ B = {x | x ∈ A or x ∈ B}.
- Intersection ( ∩ ): Identifies common elements, defined as A ∩ B = {x | x ∈ A and x ∈ B}.
- Difference ( \ ): Elements in one set that aren’t in another, noted as A \ B = {x | x ∈ A and x ∉ B}.
- Complement: Contains all elements outside a set; for set A, A' = U \ A where U is the universal set.
- Power Set: Comprises all possible subsets of a set A, including the empty set and A itself, denoted as P(A).
- Cardinality measures the number of elements in a set; finite sets have finite cardinality, while infinite sets have infinite cardinality.

Graphs are mathematical structures consisting of:
- A set of vertices (V) and a set of edges (E) connecting these vertices.
Types of Graphs include:
- Undirected Graph: Edges without direction, indicating mutual relationships.
- Directed Graph (Digraph): Edges that have directional arrows indicating a one-way relationship.
Basic Properties of Graphs involve:
- Degree: Counts the number of edges incident to a vertex.
- Path: A sequence connecting vertices through edges.
- Cycle: A path that starts and ends at the same vertex, not reusing any edges.
Combinatorial Techniques in graph theory:
- Counting Paths utilizes methods like the multiplication principle where total possibilities arise from sequential choices.
- Subgraphs are smaller graphs made from selected vertices and edges of a larger graph.
- Graph Coloring assigns distinct colors to vertices to ensure no adjacent vertices share the same color; useful in scheduling and mapping tasks.
Applications of Graph Theory span areas such as network design, resource distribution, and social network analysis.