Podcast
Questions and Answers
What is a proposition?
What is a proposition?
- An assumption made in an argument
- A statement which is either true or false (correct)
- A conclusion reached after reasoning
- A question that requires an answer
What is a simple proposition?
What is a simple proposition?
A proposition that cannot be split into smaller propositions
What is a compound proposition?
What is a compound proposition?
A proposition that has been built from two propositions using a connective
What is a connective?
What is a connective?
What is the symbol for 'and'?
What is the symbol for 'and'?
What is a tautology?
What is a tautology?
What is a contradiction?
What is a contradiction?
What is a contingent proposition?
What is a contingent proposition?
What does logical equivalence mean?
What does logical equivalence mean?
What is a theorem?
What is a theorem?
What is the Law of Syllogism?
What is the Law of Syllogism?
What is proof by contradiction?
What is proof by contradiction?
What is the well-ordering property?
What is the well-ordering property?
What defines an isomorphism of graphs?
What defines an isomorphism of graphs?
What is a complete graph?
What is a complete graph?
What is the degree of a vertex?
What is the degree of a vertex?
What is a Hamilton Path/Circuit?
What is a Hamilton Path/Circuit?
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Study Notes
Propositions
- A proposition is a statement that can be classified as either true or false.
- Simple propositions are indivisible and cannot be broken down into smaller propositions.
- Compound propositions result from combining two or more propositions using logical connectives.
Connectives
- Connectives are symbols like "and" (Λ), "or" (V), and "implies" (→) that relate propositions.
- Truth tables illustrate how propositions interact under different connectives.
Truth Tables
- The truth table for "and" (Λ) shows that it is only true when both propositions are true.
- The truth table for "or" (V) indicates it is true when at least one proposition is true.
- The "implies" (→) truth table showcases a false result only when the first proposition is true and the second is false.
- "If and only if" (↔) shows equivalency, being true only when both propositions share the same truth value.
Types of Propositions
- A tautology is a proposition that is always true.
- A contradiction is a proposition that is always false.
- A contingent proposition can vary between true and false based on its components.
Logical Relationships
- Logical Equivalence occurs if two propositions yield the same truth value across all scenarios.
- Logical Implication denotes that if one proposition is true, the other must also be true if their implication form is a tautology.
Theorems and Proofs
- A theorem is a proposition that has been proven as true under specified conditions.
- A hypothesis is a starting assumption for reaching a conclusion in logical reasoning.
- Proofs can take various forms, including direct proof, proof by contradiction, proof by construction, and proof by induction.
Mathematical Structures
- Natural Numbers (N) encompass all non-negative integers starting from 0.
- Integers (Z) comprise all whole numbers, both positive and negative.
- Prime numbers are whole numbers with exactly two distinct positive factors: 1 and themselves.
Algorithms and Correctness
- An algorithm is deemed correct if it successfully terminates and fulfills its purpose.
- The Euclidean Algorithm helps determine the greatest common divisor (gcd) of two integers through iterative division.
Graph Theory Basics
- A graph consists of vertices (nodes) and edges (connections between nodes).
- Simple graphs do not include multiple edges or loops, while multigraphs may allow multiple edges between the same vertices.
- Directed graphs have edges with direction, whereas undirected graphs treat edges as bidirectional.
Special Graphs
- A complete graph is one where every vertex is connected to every other vertex.
- A bipartite graph contains two distinct sets of vertices with edges only connecting between these sets.
- The handshaking theorem states that the sum of degrees of all vertices is twice the number of edges in the graph.
Euler and Hamiltonian Paths
- An Euler path visits every edge exactly once; an Euler circuit returns to the start vertex.
- A Hamiltonian path visits every vertex exactly once, while a Hamiltonian circuit does so, returning to the starting point.
Set Theory Concepts
- A set is an unordered collection of distinct elements, while an ordered tuple maintains element order.
- The power set of a set includes all possible subsets, with a size of 2^n, where n is the number of elements in the original set.
Relations and Properties
- A binary relation is a subset of the Cartesian product of two sets, defining relationships between elements.
- Reflexivity, symmetry, and transitivity are properties important for classifying relations.
- An equivalence relation is reflexive, symmetric, and transitive, leading to the notion of equivalence classes.
Partitions and Orderings
- A partition of a set divides it into disjoint subsets that cover all elements without overlap.
- A partial ordering (or poset) is a relation that is reflexive, anti-symmetric, and transitive, enabling a structured hierarchy among elements.
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