1. Functions

2. L62 Agenda Section 1.8: Functions Domain, co-domain, range Image, pre-image One-to-one, onto, bijective, inverse Functional composition and exponentiation Ceiling “  ” and floor “  ”

3. L63 Functions In high-school, functions are often identified with the formulas that define them. EG: f (x ) = x 2 This point of view does not suffice in Discrete Math. In discrete math, functions are not necessarily defined over the real numbers. EG: f (x ) = 1 if x is odd, and 0 if x is even. So in addition to specifying the formula one needs to define the set of elements which are acceptable as inputs, and the set of elements into which the function outputs.

4. L64 Functions. Basic-Terms. DEF: A function f : A  B is given by a domain set A, a codomain set B, and a rule which for every element a of A, specifies a unique element f (a) in B. f (a) is called the image of a, while a is called the pre-image of f (a). The range (or image) of f is defined by f (A) = {f (a) | a  A }.

5. L65 Functions. Basic-Terms. EG: Let f : Z  R be given by f (x ) = x 2 Q1: What are the domain and co-domain? Q2: What’s the image of -3 ? Q3: What are the pre-images of 3, 4? Q4: What is the range f (Z) ?

6. L66 Functions. Basic-Terms. f : Z  R is given by f (x ) = x 2 A1: domain is Z, co-domain is R A2: image of -3 = f (-3) = 9 A3: pre-images of 3: none as  3 isn’t an integer! pre-images of 4: -2 and 2 A4: range is the set of perfect squares f (Z) = {0,1,4,9,16,25,…}

7. L67 Functions. Sub-ranges. The effect of functions on subsets of the domain is often important. DEF: Given a function f : A  B. The pre- image set (or inverse image) of b is defined by f -1 (b) = {a  A | f (a)=b }. Given subsets S  A and T  B, the image set of S is defined by f (S ) = {f(a ) | a  S } and the pre-image set (or inverse image) of T is defined by f -1 (T ) = {a  A | f (a)  T }. NOTE: Even when f is not invertible, the inverse image is defined!

8. L68 Functions. Sub-ranges. EG: f : Z  R with f (x ) = x 2 Q1: Calculate f –1 (3) Q2: Calculate f –1 (4) Q3: Calculate f ( {-9,-5,-3,0,1,2,3,4} ) Q4: Calculate f –1 ({-9,-5,-3,0,0.25,1,2,2.25,3,4})

9. L69 Functions. Sub-ranges. EG: f : Z  R with f (x ) = x 2 A1: f –1 (3) =  A2: f –1 (4) = {-2, 2} A3: f ( {-9,-5,-3,0,1,2,3,4} ) = {81,25,9,0,1,4,16} A4:f –1 ({-9,-5,-3,0,0.25,1,2,2.25,3,4}) = {0,-1,1,-2,2}

10. L610 One-to-One, Onto, Bijection. Intuitively. Represent functions using “node and arrow” notation: One-to-One means that no clashes occur. BAD:a clash occurred, not 1-to-1 GOOD:no clashes, is 1-to-1 Onto means that every possible output is hit BAD: 3 rd output missed, not onto GOOD:everything hit, onto

11. L611 One-to-One, Onto, Bijection. Intuitively. Bijection means that when arrows reversed, a function results. Equivalently, that both one-to-one’ness and onto’ness occur. BAD:not 1-to-1. Reverse over-determined: BAD:not onto. Reverse under-determined: GOOD:Bijection. Reverse is a function:

12. L612 One-to-One, Onto, Bijection. Formal Definition. DEF: A function f : A  B is: one-to-one (or injective) if different elements of A always result in different images in B. onto (or surjective) if every element in B is hit by f. I.e., f (A ) = B. a one-to-one correspondence (or a bijection, or invertible) if f is both one-to-one as well as onto. If f is invertible, its inverse f -1 : B  A is well defined by taking the unique element in the pre- image of b, for each b  B.

13. L613 One-to-One, Onto, Bijection. Examples. Q: Which of the following are 1-to-1, onto, a bijection? If f is invertible, what is its inverse? 1. f : Z  R is given by f (x ) = x 2 2. f : Z  R is given by f (x ) = 2x 3. f : R  R is given by f (x ) = x 3 4. f : Z  N is given by f (x ) = |x |

14. L614 One-to-One, Onto, Bijection. Examples. 1. f : Z  R, f (x ) = x 2 : none 1. not 1-1 clashes for -1,1 in Z 2. not onto -1,-2 missed from R 2. f : Z  R, f (x ) = 2x : 1-1 3. f : R  R, f (x ) = x 3 : 1-1, onto, bijection, inverse is f (x ) = x (1/3) 4. f : Z  N, f (x ) = |x |: onto

15. L615 Composition When a function f spits out elements of the same kind that another function g eats, f and g may be composed by letting g immediately eat each output of f. DEF: Suppose that g : A  B and f : B  C are functions. Then the composite f  g : A  C is defined by setting f  g (a) = f ( g (a) )

16. L616 Composition. Examples. Q: Compute g  f where 1.f : Z  R, f (x ) = x 2 and g : R  R, g (x ) = x 3 2. f : Z  Z, f (x ) = x + 1 and g = f -1 so g (x ) = x – 1 3. f : {people}  {people}, f (x ) = the father of x, and g = f

17. L617 Composition. Examples. 1.f : Z  R, f (x ) = x 2 and g : R  R, g (x ) = x 3 f  g : Z  R, f  g (x ) = x 6 2. f : Z  Z, f (x ) = x + 1 and g = f -1 f  g (x ) = x (true for any function composed with its inverse) 3. f : {people}  {people}, f (x ) = g(x ) = the father of x f  g (x ) = grandfather of x from father’s side

18. L618 Repeated Composition When the domain and codomain are equal, a function may be self composed. The composition may be repeated as much as desired resulting in functional exponentiation. The whole process is denoted by f n (x ) = f  f  f  f  …  f (x ) where f appears n –times on the right side. Q1: Given f : Z  Z, f (x ) = x 2 find f 4 Q2: Given g : Z  Z, g (x ) = x + 1 find g n Q3: Given h(x ) = the father of x, find h n

19. L619 Repeated Composition A1: f : Z  Z, f (x ) = x 2. f 4 (x ) = x (2*2*2*2) = x 16 A2: g : Z  Z, g (x ) = x + 1 g n (x ) = x + n A3: h (x ) = the father of x, h n (x ) = x ’s n’th patrilineal ancestor

20. L620 Ceiling and Floor This being a course on discrete math, it is often useful to discretize numbers, sets and functions. For this purpose the ceiling and floor functions come in handy. DEF: Given a real number x : The floor of x is the biggest integer which is smaller or equal to x The ceiling of x is the smallest integer greater or equal to x. NOTATION: floor(x) =  x , ceiling(x) =  x  Q: Compute  1.7 ,  -1.7 ,  1.7 ,  -1.7 .

21. L621 Ceiling and Floor A:  1.7  = 1,  -1.7  = -2,  1.7  = 2,  -1.7  = -1 Prove : show that for all positive real numbers x, y:  x.y  <=  x .  y 