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课程名称：离散数学 - 集合、关系与函数 课程概述： 本课程涵盖集合、关系和函数的基本概念与运算。主要内容包括： 1. **集合**：定义及其表示法；空集合；有限集合与无限集合；相等集合与子集；实数集合的子集，特别是区间（带符号表示）；权力集合；普遍集合；维恩图；集合的并、交、差及补集；补集的性质；基于集合的实际问题。 2. **关系与函数**：有序对；集合的笛卡尔积；有限集合笛卡尔积的元素数量；实数的笛卡尔积；关系的定义；图示化图形；关系的定义域、余域及值域；作为特殊关系的函数；函数的图示表示；实值函数及其定义域、值域，包括常数函数、恒等函数、多项式、分式、绝对值、符号函数、指数函数、对数函数、最大整数函数（包含其图形）；函数的和、差、乘积和商等运算。 3. **总结**： - 集合的基本定义和运算，包括空集合、有限与无限集合、子集的定义，以及集合之间的基本运算（并、交、差、补等）的性质和实际应用。 - 关系的各种类型（如空关系、普遍关系、反射关系、对称关系、传递关系和等价关系）及其特性与定义。 - 函数的定义及其各种特性（如单射、满射、双射），函数的复合及可逆性。 适用人群：本课程适合成人学习。
SetsSets and their representationsEmpty setFinite and Infinite setsEqual sets. SubsetsSubsets of a set of real numbers especially intervals (with notations)Power setUniversal setVenn diagramsUnion and Intersection of setsDifference of setsComplement of a setProperties of Complement SetsPractical Problems based on setsRelations & FunctionsOrdered pairsCartesian product of setsNumber of elements in the cartesian product of two finite setsCartesian product of the sets of real (up to R × R)Definition of −RelationPictorial diagramsDomainCo-domainRange of a relationFunction as a special kind of relation from one set to anotherPictorial representation of a function, domain, co-domain and range of a functionReal valued functions, domain and range of these functions −ConstantIdentityPolynomialRationalModulusSignumExponentialLogarithmicGreatest integer functions (with their graphs)Sum, difference, product and quotients of functionsSUMMARYSets - This chapter deals with some basic definitions and operations involving sets. These are summarised below: 1. A set is a well-defined collection of objects. A set which does not contain any element is called empty set. 2. A set which consists of a definite number of elements is called finite set, otherwise, the set is called infinite set. 3. Two sets A and B are said to be equal if they have exactly the same elements. 4. A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of R. 5. A power set of a set A is collection of all subsets of A. It is denoted by P(A).6. The union of two sets A and B is the set of all those elements which are either in A or in B. 7. The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B. 8. The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A. 9. For any two sets A and B, (A ∪ B)′ = A′ ∩ B′ and ( A ∩ B )′ = A′ ∪ B′ 10. If A and B are finite sets such that A ∩ B = φ, then n (A ∪ B) = n (A) + n (B). If A ∩ B ≠ φ, then n (A ∪ B) = n (A) + n (B) - n (A ∩ B)Relations & Functions - In this chapter, we studied different types of relations and equivalence relation, composition of functions, invertible functions and binary operations. The main features of this chapter are as follows: 1. Empty relation is the relation R in X given by R = φ ⊂ X × X. 2. Universal relation is the relation R in X given by R = X × X. 3. Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X. 4. Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R. 5. Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R. 5. Equivalence relation R in X is a relation which is reflexive, symmetric and transitive. 6. Equivalence class [a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.7. A function f: X → Y is one-one (or injective) if f(x1 ) = f(x2 ) ⇒ x1 = x2 ∀ x1 , x2 ∈ X. 8. A function f: X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y. 9. A function f: X → Y is one-one and onto (or bijective), if f is both one-one and onto. 10. The composition of functions f: A → B and g: B → C is the function gof: A → C given by gof(x) = g(f(x)) ∀ x ∈ A. 11. A function f: X → Y is invertible if ∃ g: Y → X such that gof = IX and fog = IY. 12. A function f: X → Y is invertible if and only if f is one-one and onto."This course is for Adults"
