4) Have you met any functions during our previous lectures in this course?

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1 Definition: Let X and Y be sets. A function f from the set X to the set Y is a rule which associates to each element x X a unique element y Y. Notation: f : X Y f defined on X with values in Y. x y y = f(x). Examples: f : { children under } { women } ) x f(x) = x s mother. Each child has exactly one biological mother, so for each x there exists a unique f(x). ) f : { people on Earth } { locations on Earth } x f(x) = location of x now. You can t be in two different locations at the same time, but you ve got to be somewhere, so for each x there exists a unique f(x). ) What are some of the first functions you have encountered? ) Have you met any functions during our previous lectures in this course?

2 Definition: The graph of a function f : X Y is the subset of X Y defined as follows: Graph (f) := {(x,f(x)) X Y; x X}. Examples: )Consider A = {a,c}, the universal set U = {a,b,c,d} and c A : U {0,} its characteristic function. x U a b c d c A (x) )Let A = {a,c} and f : P(A) N given by f(b) = B for all B A. B P(A) f(b)

3 If f : R R, then Graph(f) is a set of points in the plane R R. Example: f : R R given by f(x) = { x if x < x +8x if x Note: The Graph(f) R R helps you find f(x) for any x: Find x on the Ox axis. Draw a vertical line through it. It will meet the Graph(f) at a unique point: (x, f(x)). The horizontal line through (x,f(x)) meets the Oy axis at the point y = f(x).

4 Notations: f : X Y function. X = the domain of f, Y = the codomain of f, f(x) = {f(x) ; x X} = the range of f. Examples: For each of the following functions, choose suitable domains, codomain and find its range. ) f(x) = e x. ) f(x) = lnx. ) f(x) = x +x.

5 Composition of functions Definition: Given f : X Y and g : Y Z, define their composition g f : X Z by (g f)(x) = g(f(x)). Example: f : R (,] and g : (,] R, { x if x < f(x) = x +8x if x (g f)(x) = g(x) = x. (f g)(x) =

6 Functions and counting. Are there more students, or more chairs in the classroom?. Are there more books or more shelves in Boole library?. Are there more women, or more children under in the world? Consider the following functions: c : { students in this classroom } { chairs in this classroom }, x c(x) = x s chair. b : { books in Boole library } { shelves in Boole library}, x b(x) = the shelf on which x lies. m : { children on Earth under } { women on Earth}, x m(x) = x s biological mother. Explain : Explain : Explain :

7 Injective (One-to-One) Functions f : X Y Equivalent definitions for an injective function: f is injective f is injective f is injective Examples of injective functions: Equivalent definitions for a NON injective function: f is NOT injective f is NOT injective f is NOT injective Examples of NON injective functions:

8 Surjective (Onto) Functions f : X Y Equivalent definitions for a surjective function: f is surjective f is surjective f is surjective Examples of surjective functions: Equivalent definitions for a NON surjective function: f is NOT surjective f is NOT surjective f is NOT surjective Examples of NON surjective functions:

9 For each of these functions: is it injective? surjective?

10 Bijective Functions (One-to-One Correspondences) f : X Y Definition: A function f : X Y is called bijective it is both injective and surjective. Definition: A function f : X Y is called invertible there exists another function f : Y X such that (f f)(x) = x, x X and (f f )(y) = y, y Y. Then f is called the inverse of f. Proposition: A function f : X Y is bijective it is invertible. Proof: f : X Y is bijective y Y,!x X such that f(x) = y. Let f (y) := x. Thus we get f : Y X satisfying: (f f)(x) = f (f(x)) = f (y) = x, (f f )(y) = f(f (y)) = f(x) = y. Definition: Two sets X and Y are said to have the same cardinality if there exists a bijective function f : X Y. Definition: Any set having the same cardinality with N is called countable.

11 Which of Z, Q, R, C are countable sets?. Is Z countable? z Z n N Is Q countable? Q = {q Q; q > 0} {q Q; q 0}. p/q

12 . Is R countable? a) Let f : (0,) (0, ), f(x) = x x. Prove f is a bijection: b) Can you find a bijection g : (0, ) R? c) Construct a bijection R (0,). Thus R and (0,) have the same cardinality.

13 . Is (0,) countable? NO! Proof: Cantor s diagonal argument. Assume that (0,) can be written as a sequence of numbers r,r,r,r,... Each number r n = 0.r n r n r n r n r n... For example, r n = 0. π has r n =,r n =,r n =,r n =,r n =,... Construct a table out of the numbers r ni : st nd rd th...decimals r r r r r... r ni r r r r r... r r r r r... r r r r r Consider the number x = 0.s s s s... formed by modifying the elements on the diagonal: s i := r ii + for all i, except for r ii = 9 when s i := 0. x (0,) so x = r k for some k. However, by construction, x s k-th decimal is Contradiction! s k = r kk + r kk, or s k = 0 r kk when r kk = 9. Thus (0,) is not countable and so R is not countable.