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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/05%3A_Graph_Theory/5.04%3A_PathsThis page discusses key concepts in graph theory, including definitions of walks, trails, and paths, highlighting their distinct characteristics regarding vertex and edge repetition. It explains Euler...This page discusses key concepts in graph theory, including definitions of walks, trails, and paths, highlighting their distinct characteristics regarding vertex and edge repetition. It explains Eulerian trails and Hamiltonian paths and addresses graph connectivity, stating that a graph is connected if any vertex pair has a connecting path. Additionally, it introduces n-connected graphs and includes practice checkpoints to reinforce understanding of these concepts.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/05%3A_Graph_Theory/5.05%3A_CyclesThis page defines important graph theory terms such as circuit, cycle, Eulerian circuit, and Hamiltonian circuit. It explains that a circuit is a closed walk with the same starting and ending vertex, ...This page defines important graph theory terms such as circuit, cycle, Eulerian circuit, and Hamiltonian circuit. It explains that a circuit is a closed walk with the same starting and ending vertex, while a cycle does not repeat vertices. An Eulerian circuit covers every edge once, and a Hamiltonian circuit visits each vertex once. Additionally, it offers practice checkpoints for readers to engage with the concepts, including drawing cycles and identifying specific types of circuits.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/05%3A_Graph_Theory/5.02%3A_Properties_of_GraphsThis page provides definitions and examples of graph properties like adjacency, vertex degrees, and types of graphs (regular, complete, bipartite). It covers subgraphs, graph complements, and duals, a...This page provides definitions and examples of graph properties like adjacency, vertex degrees, and types of graphs (regular, complete, bipartite). It covers subgraphs, graph complements, and duals, along with practice checkpoints for calculating degrees and understanding independent sets and maximum matchings. Each definition is illustrated with examples to aid in the comprehension of graph theory concepts.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/01%3A_Functions_and_Graphs/1.03%3A_Trigonometric_Functions/1.3E%3A_Exercises_for_Section_1.3This page contains exercises on angle conversions, trigonometric functions, triangle side calculations, and solving identities. It covers sine and cosine equations, including amplitude, period, and ph...This page contains exercises on angle conversions, trigonometric functions, triangle side calculations, and solving identities. It covers sine and cosine equations, including amplitude, period, and phase shifts, along with applications in geometry, angular speed, and natural phenomena models. Additionally, it features mathematical models predicting temperature and tide height with their respective amplitudes and periods, complemented by graphs illustrating their periodic variations.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.03%3A_Equivalence_RelationsThe main idea of an equivalence relation is that it is something like equality, but not quite. Usually there is some property that we can name, so that equivalent things share that property. For examp...The main idea of an equivalence relation is that it is something like equality, but not quite. Usually there is some property that we can name, so that equivalent things share that property. For example Albert Einstein and Adolf Eichmann were two entirely different human beings, if you consider all the different criteria that one can use to distinguish human beings there is little they have in common.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/11%3A_Appendix_-_Prerequisite_Function_Material/11.02%3A_RelationsThis section introduces relations, explaining how they are defined as sets of ordered pairs. It provides examples of different types of relations and discusses how to represent relations graphically a...This section introduces relations, explaining how they are defined as sets of ordered pairs. It provides examples of different types of relations and discusses how to represent relations graphically and algebraically.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Yet_Another_Introductory_Number_Theory_Textbook_-_Cryptology_Emphasis_(Poritz)/04%3A_CryptologyBeware that cryptography is widely (but inappropriately!) used as a synecdoche for cryptology. (This is not unlike the widely understood incorrect usage of the word hacker.) We will try to use these w...Beware that cryptography is widely (but inappropriately!) used as a synecdoche for cryptology. (This is not unlike the widely understood incorrect usage of the word hacker.) We will try to use these words more carefully. There will be very little number theory, but we will set up some terminology and simple examples of cryptography and the corresponding cryptanalysis, with an emphasis on the old, historic, systems which are no longer viable in the modern age.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370%3A_Precalculus/01%3A_Relations_and_Functions/1.02%3A_RelationsThis section introduces relations, explaining how they are defined as sets of ordered pairs. It provides examples of different types of relations and discusses how to represent relations graphically a...This section introduces relations, explaining how they are defined as sets of ordered pairs. It provides examples of different types of relations and discusses how to represent relations graphically and algebraically.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/03%3A_Graph_Theory/11%3A_Basics_of_Graph_Theory/11.02%3A_Basic_Definitions_Terminology_and_NotationNow that we have an intuitive understanding of what a graph is, it is time to make a formal definition.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.01%3A_Systems_of_Linear_Equations/1.1E%3A_Exercises_for_Section_1.1This page offers exercises on solving linear systems graphically, focusing on finding intersection points of lines, understanding different solution scenarios (no, unique, infinite solutions), and exa...This page offers exercises on solving linear systems graphically, focusing on finding intersection points of lines, understanding different solution scenarios (no, unique, infinite solutions), and examining common intersections of multiple lines or planes. It includes a word problem involving weights of four individuals and tasks requiring the construction of linear systems with defined solution properties and graphical relationships.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/01%3A_Integration/1.08%3A_Chapter_1_Review_ExercisesThis page features calculus exercises on definite integrals, Riemann sums, and antiderivatives. It includes exercises on evaluating mathematical truths and real-world applications, such as calculating...This page features calculus exercises on definite integrals, Riemann sums, and antiderivatives. It includes exercises on evaluating mathematical truths and real-world applications, such as calculating average costs and velocities. The content ranges from theoretical proofs to practical scenarios, emphasizing the continuity of functions and derivatives. Specific calculations and their answers are provided, demonstrating the connections between theory and application.
