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수학/이산수학

[Discrete Mathematics] lecture 8 - Graphs IntroductionGraph is a particular class of discrete structure for representing relations. We deal with various types of graphs, their representations and trees.GraphsDefinition 1A simple graph $G = \left$ consists of‣ a set $V$ of vertices(=nodes)‣ a set $E$ of edges(=arcs, links), which are unordered pairs of distinct elements in $V$. Definition 2A multigraph is a graph with multiple edges. For.. 더보기
[Discrete Mathematics] lecture 7 - lattice OverviewThis lecture covers the lattice theory and boolean algebra.Introduction to LatticeDefinition 1 (lattice) A poset $\left$ which every pair of elements in L has lub and glb is called a lattice.lub is denoted as $+$ and glb is denoted as $*$.We say that $a$ and $b$ join at $a+b$ and meet at $a*b$.Example 1.Consider a poset $\left$ where $L = {a, b, c, d}$ and $ \preceq = I_L ∪ {(a, c),(a, d.. 더보기
[Discrete Mathematics] lecture 6 - algebra OverviewThis lecture introduces the concept of an algebraic structure, defined by a carrier set, operations, and constants. It covers foundational ideas like closure, subalgebras, identity and zero elements, and inverses. Key structures are introduced, including semigroups, monoids, and groups, along with the hierarchy among them.The lecture also defines homomorphisms (structure-preserving maps .. 더보기
[Discrete Mathematics] lecture 5 - counting OverviewThis lecture introduces the fundamentals of counting and recurrence relations. It begins by defining finite and infinite sets, emphasizing the concept of cardinality and distinguishing between countable and uncountable sets. Key counting principles such as the Pigeonhole Principle, sum and product rules, and formulas for permutations and combinations are presented, along with proofs and .. 더보기
[Discrete Mathematics] lecture 4 - Functions OverviewThis lecture covers the basics of functions in discrete mathematics, defining them as relations where each input has a unique output. Key terms like domain, codomain, and range are introduced. It explores composite functions, operators, restrictions, and extensions.Different types of functions are discussed: injective (one-to-one), surjective (onto), bijective (both), along with their pr.. 더보기
[Discrete Mathematics] lecture 3 - Relations OverviewThis lecture presents a comprehensive introduction to relations in discrete mathematics, covering foundational definitions, properties, and theorems. It begins by defining binary relations and their operations (complement, inverse, composition, and power), followed by a detailed look at key relational properties such as reflexivity, symmetry, and transitivity. The text explores the graph.. 더보기
[Discrete Mathematics] lecture 1 - Basics of Logic * 원본 포스팅은 여기를 참조해주세요What is logic?A formal system for describing knowledge and implementing reasoning on knowledge.Logic is just like a language. But as it is used for reasoning, it eleminates ambiguity.Just like a language, it consistof syntax and semantics. Syntax is the rules for constructing sentences, and semantics is the meaning of the sentences. But in logic, there is a set of rules for d.. 더보기

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