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Questions and Answers
What is a proposition?
What is a simple proposition?
A proposition that cannot be split into smaller propositions
What is a compound proposition?
A proposition that has been built from two propositions using a connective
What is a connective?
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What is the symbol for 'and'?
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What is a tautology?
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What is a contradiction?
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What is a contingent proposition?
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What does logical equivalence mean?
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What is a theorem?
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What is the Law of Syllogism?
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What is proof by contradiction?
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What is the well-ordering property?
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What defines an isomorphism of graphs?
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What is a complete graph?
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What is the degree of a vertex?
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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" (Î›), "or" (V), and "implies" (â†’) that relate propositions.
- Truth tables illustrate how propositions interact under different connectives.
Truth Tables
- The truth table for "and" (Î›) 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" (â†’) truth table showcases a false result only when the first proposition is true and the second is false.
- "If and only if" (â†”) 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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Description
Test your knowledge of propositional logic with these flashcards from COMPSCI 225. Each card defines key terms such as propositions, simple propositions, compound propositions, and connectives. Perfect for reinforcing your understanding and preparing for exams.