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

2. Definition A set is a collection of objects. A set is a collection of objects. The objects in a set are called elements of the sets. 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. 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) Z: integers (positive, negative, zero) N: positive integers or natural numbers (not including zero) N: positive integers or natural numbers (not including zero) Q: rational numbers Q: rational numbers R: real number R: real number C: complex numbers C: complex numbers  : there exists  : there exists  : for all  : 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 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 A is called the domain of f Set B is called the codomain of f Set B is called the codomain of f f[A] is called the image of the mapping 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 x 1, x 2  A, f(x 1 )=f(x 2 )  x 1 =x 2 or if x 1  x 2  f(x 1 )  f(x 2 )

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

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

16. Increasing function f is said to be monotonic increasing in (a,b) if and only if f(x 1 )  f(x 2 )  b > x 1 > x 2 > a. f is said to be monotonic increasing in (a,b) if and only if f(x 1 )  f(x 2 )  b > x 1 > x 2 > a. f is said to be strictly increasing in (a,b) if and only if f(x 1 ) > f(x 2 )  b > x 1 > x 2 > a. f is said to be strictly increasing in (a,b) if and only if f(x 1 ) > f(x 2 )  b > x 1 > x 2 > a.

17. Decreasing function f is said to be monotonic decreasing in (a,b) if and only if f(x 1 )  f(x 2 )  b > x 1 > x 2 > a. f is said to be monotonic decreasing in (a,b) if and only if f(x 1 )  f(x 2 )  b > x 1 > x 2 > a. f is said to be strictly decreasing in (a,b) if and only if f(x 1 ) x 1 > x 2 > a. f is said to be strictly decreasing in (a,b) if and only if f(x 1 ) x 1 > x 2 > 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.