outline of discrete mathematics and its applications(2.1-2.3)

2.1 sets

• definition
• ways to express
  
  • roster method
  • set builder notation
  • interval notation
• singletonset(单元素集)
• venn diagrams(维恩图)
• subsets
  
  • subsets definition
  • ∀x(x∈A->x∈B)
• the size of a set
  
  • finite set(有限集)
  • infinite set(无限集)
  • cardinality(基数)
  • power set P(S) : the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S).
• Cartesian Products
  
  • ordered collection
  • Let A and B be sets. The Cartesian product of A and B denoted by A×B, is the set of all orderer pairs, where a∈A and b∈B
  • A×B = {(a,b)|a∈A∧b∈B}
  • the Cartesian product of the sets A1, A2, A3. …, An
• using set notation with quantifiers
• truth sets and quantifiers

2.2 set operations

• A∪B
• A∩B
• two sets are called disjoint if their intersection is the empty set.
• difference of A and B A-B={x|x∈A∧x∉B}
• complement of A with respect to U(A关于U的补集)
• set indentities(好多laws不再列举)(这些规则用逻辑演算的方法可以证明，看例子)
• generalized unions and intersections
• computer representation of sets

2.3 functions

• definition of function
• domain(定义域) codomain(陪域:比值域要大)
• preimage(原象) range or image
• if f is a function from A to B we say that f maps A to B
• One-to-One(单射) Onto(满射 surjection)
• bijection(双射)
• increasing strictly increasing
• decreasing strictly decreasing
• inverse function(反函数)
• 复合函数
• the graphs of functions
  
  • floor function
  • ceiling function
• partial functions