1 / 19

Sets and Functions

Sets and Functions. Contents Set language Basic knowledge on sets Intervals Functions (Mappings). Definition . A set is a collection of objects. The objects in a set are called elements of the sets. Symbol. e.g. S ={a,b,c} is a set and a , b , c are elements.

Download Presentation

Sets and Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Sets and Functions Contents Set language Basic knowledge on sets Intervals Functions (Mappings)

  2. Definition • A set is a collection of objects. • The objects in a set are called elements of the sets.

  3. Symbol e.g. S ={a,b,c} is a set and a, b, c are elements. aS means a belongs to S or a is an element of S, otherwise, we write a S.

  4. Standard notation • Z: integers (positive, negative, zero) • N: positive integers or natural numbers (not including zero) • Q: rational numbers • R: real number • C: complex numbers • : there exists • : for all

  5. Equality of sets A=B if and only if for any x, x  A  x  B

  6. Subsets(子集) A is a subset of B, written A  B, if and only if for any x, x  A  x  B Note: A  A, A is an improper subset of itself.

  7. The empty set(空集) The empty set, denoted by , is a set which contains no elements.

  8. Union of sets(倂集) The union of two sets A and B is defined as the set A  B = {x: x  A or x  B}

  9. Intersection of sets(交集) The intersection of two sets A and B is defined as the set A  B = {x: x  A and x  B}

  10. Intervals open interval: x  (a,b) means a < x < b closed interval: x  [a,b] means a  x  b

  11. Functions函數(Mappings映射) f: A  B • Set A is called the domain of f • Set B is called the codomain of f • f[A] is called the image of the mapping f

  12. Surjective (onto)(滿射) f: AB If f [A] = B, then f is a surjective function (mapping). i.e.  y  B,  x  A such that f(x)=y

  13. Injective (one-to-one)(內射) f: AB f is injective if each element of B is the image of at most one element of A. i.e. for some x1, x2 A, f(x1)=f(x2)  x1=x2 or if x1x2  f(x1)f(x2)

  14. Bijective (one-to-one correspondence)(雙射) If f is both surjective and injective, then f is bijective

  15. Well-defined • Constant function • Identity function(恆等函數) • Composite function(複合函數) • Inverse function(逆像)

  16. Increasing function • f is said to be monotonic increasing in (a,b) if and only iff(x1)  f(x2)  b > x1 >x2 > a. • f is said to be strictly increasing in (a,b) if and only iff(x1) > f(x2)  b > x1 >x2 > a.

  17. Decreasing function • f is said to be monotonic decreasing in (a,b) if and only iff(x1)  f(x2)  b > x1 >x2 > a. • f is said to be strictly decreasing in (a,b) if and only if f(x1) < f(x2)  b > x1 >x2 > a.

  18. Periodic function A function is said to be periodic , with period of if and only if f(x+) = f(x)  x  R

  19. Bounded(有界) A function is said to be bounded (有界) on an interval I if there is a positive number M such that |f(x)| M for any x  I.

More Related