Sets and Functions

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.