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Questions and Answers
What does 'Discrete Structures' refer to?
What does 'Discrete Structures' refer to?
A branch of mathematics that studies objects that can be counted, are finite and distinct from one another.
Which of these represents a Set of natural numbers?
Which of these represents a Set of natural numbers?
What is a graph in graph theory?
What is a graph in graph theory?
A data structure consisting of nodes (vertices) connected by edges.
What type of graph has edges that have no direction?
What type of graph has edges that have no direction?
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Define a weighted graph.
Define a weighted graph.
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What is a loop in graph theory?
What is a loop in graph theory?
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What does the term 'degree of a graph' refer to?
What does the term 'degree of a graph' refer to?
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What is the indegree of a vertex?
What is the indegree of a vertex?
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What is the outdegree of a vertex?
What is the outdegree of a vertex?
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Study Notes
Discrete Structures
- Discrete structures is a branch of mathematics that studies countable, finite, and distinct objects.
- These structures are fundamental in computer science.
- Examples include sets, graph theory, and trees.
Sets
- A set is a collection of finite and distinct objects.
- Natural numbers: {1, 2, 3...}
- Set of courses in semester 2: {200P, DLD, DS}
Graph Theory
- A graph is a data structure consisting of nodes (also called vertices) connected by edges.
- Graphs are used to represent relationships between objects.
Types of Graphs
- Directed Graph: A graph in which the direction of an edge is defined to a particular node. Arrows indicate direction.
- Undirected Graph: A graph in which edges have no direction.
Complete Graph
- A graph in which each vertex is connected to every other vertex.
- Example: Tournament graph where every player plays every other player
Weighted Graphs
- A graph in which edges have weights or costs associated with them.
- Weights can represent distances between cities in a road network.
Loops in Graphs
- A loop (also called a self-loop or buckle) is an edge that connects a vertex to itself.
Degree of a Graph
- The degree of a vertex is the number of vertices adjacent to a vertex.
Degree of Undirected Graph
- deg(a) = 2, deg (d) = 2
- deg(b) = 3, deg(e) = 0 (Isolated vertex)
- deg(c) = 1 (Pendent vertex)
Degree of Directed Graph
- Indegree: The number of edges coming into a vertex.
- Outdegree: The number of edges going out from a vertex.
Adjacency Matrix
- A square matrix of size NxN where N is the number of nodes in the graph used to represent connections.
Trees
- A hierarchical structure used to represent and organize data.
- A tree is a collection of nodes connected by edges.
- It has a hierarchical relationship between nodes.
Terminologies in Trees
- Root node: The topmost node that doesn't have a parent node.
- Parent node: A node that has a direct connection to one or more child nodes.
- Child node: Nodes directly connected to a parent node.
- Siblings: Children of the same parent node.
- Leaf node: Nodes without child nodes.
- Subtree: A position or part of a tree.
Binary Trees
- A hierarchical structure where each node has at most two children (left child and right child).
Recursion
- A function that calls itself to solve a smaller part of a larger problem.
- The problem is divided into smaller sub-problems until a base case is reached, which stops the recursion.
Applications of Binary Trees
- Files compression (ZIP, JPEG, MP3)
- Scheduling systems, shortest path finding
- Organizational hierarchy
- File system directories
Permutation and Combinations
- Permutation: The arrangement of objects in a specific order.
- Combination: The selection of objects where order does not matter.
Logical Operators
- Propositions: Declarative statements that are either true or false.
- Simple Proposition: A statement that cannot be broken down into smaller statements.
- Compound Proposition: Combining two or more simple propositions using logical connectives (AND, OR, NOT).
Conditional Proposition
- Conditional Proposition: (P→Q) Implication or conditional statement.
- Antecedent / Hypothesis: Proposition 'P'.
- Consequent / Conclusion: Proposition 'Q'.
Biconditional Statement
- Biconditional Statement: (P↔Q) or (p iff q)
- A statement that is true when both p and q are true or both p and q are false.
Negation
- Negation (~P) of a proposition P is the proposition that is false when P is true and true when P is false.
Rules of Inference
- Modus Ponens: If P implies Q and P is true, then Q is true.
- Modus Tollens: If P implies Q and Q is false, then P is false.
- Hypothetical Syllogism: If P implies Q and Q implies R, then P implies R.
- Disjunctive Syllogism: If either P or Q is true and P is false, then Q is true.
- Addition: If P is true, then P or Q is true.
- Simplification: If P and Q is true, then P and Q is true.
- Constructive Dilemma: If P implies Q and R implies S and either P or R is true, then either Q or S is true.
- Destructive Dilemma: If P implies Q and R implies S and either not-Q or not-S is true, then either not-P or not-R is true..
Set Theory
- Set: A collection of well-defined and distinct objects.
- Elements: Objects belonging to a set.
- Subset: A set whose elements are all contained within another set.
- Operations on Sets: Union, intersection, difference, complement, etc..
- Universal Set(U): Contains all elements under consideration.
- Empty Set (Ø or {}): Set with no elements.
Relations
- Relation: A collection of ordered pairs, associating elements between sets.
- Domain: Set of all first elements.
- Range: Set of all second elements.
- Function: A relation where each input has exactly one output.
- Types of relations: one-to-one, many-to-one, one-to-many, many-to-many
- onto, into
Pigeonhole Principle
- If 'n' pigeonholes are occupied by more items than pigeonholes, there is at least one hole with more than one item.
- Used to determine minimum values to satisfy a constraint.
Predicate Logic
- Statement that contains variables, becomes true or false when variables are substituted.
- Useful for representing real-world scenarios.
Regular Expressions
- Sequence of characters that defines a search pattern.
- Used for matching and manipulating text.
- Examples include dot, character set, zero or more repetitions, the beginning and end of lines and strings.
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Description
Test your knowledge on discrete structures, sets, and graph theory in this engaging quiz. Challenge yourself with questions about graphs, weighted graphs, and different types of connectivity. Perfect for students looking to solidify their understanding of these mathematical concepts.
