Discrete Mathematics CS 2610 September 12, 2006

2 Agenda Last class Functions  Vertical line rule  Ordered pairs  Graphical representation  Predicates as functions This class More on functions!

3 Function Terminology Given a function f:A  B A is the domain of f. B is the codomain of f. If f(a)=b, b is the image of a under f. a is a pre-image of b under f.  In general, b may have more than 1 pre-image. The range R of f (or image of f) is : R={b |  a f(a)=b } -- the set of all images For any set S  A, the image of S,  f(S) = { b  B |  a  S, b = f(a)} For any set T  B, the inverse image of T  f −1 (T) = { a  A | f(a)  T }

4 Example Mike Mario Kim Joe Jill John Smith Edward Jones Richard Boone f AB DomainCodomain The image of Mike under f is John Smith Mike is a pre-image of John Smith under f R (f) = {John Smith, Richard Boone} f(Mike,Mario,Jill) = {John Smith, Richard Boone} f -1 (Richard Boone) = {Joe, Jill}

5 Example Given a function f: Z  Z where f(x) = x 2 -- the domain of f is the set of all integers -- the codomain of f is the set of all integers -- the range of f is the set of all integers that are perfect squares {0, 1, 4, 9, 16, 25, …}

6 Function Composition Given the functions g:A  B and f:B  C, the composition of f and g, f ○ g: A  C defined as f ○ g (a) = f ( g (a) ) h b d o  f g A BC f ○ g (h) ?

7 Function Composition Properties Associative: Given the functions g:A  B and f:B  C and h:C  D then h ○ (f ○ g)  (h ○ f ) ○ g

8 Function Self-Composition A function f: A  A (the domain and codomain are the same) can be composed with itself f: People  People where f(x) is the father of x f ○ f (Mike) is the father of the father of Mike f ○ f ○ f (Mike) ? f ○ f ○ f ○ f(Mike) ?

9 Injective Functions (one-to-one) A function f: A  B is one-to-one (injective, an injection) iff f(x) = f(y)  x = y for all x and y in the domain of f (  x  y(f(x) = f(y)  x = y)) Equivalently:  x  y(x  y  f(x)  f(y)) Every b  B has at most 1 pre-image f AB

10 Surjective Functions (onto) A function f: A  B is onto (surjective, an surjection) iff  y  x( f(x) = y) where y  B, x  A Every b  B has at least one pre-image f AB

11 Bijective Functions A function f: A  B is bijective iff it is one-to-one and onto (a one-to-one correspondence) f The domain cardinality equals the codomain cardinality A B

12 Inverse Functions Let f : A  B be a bijection, the inverse of f, f -1 :B  A such that for any b  B, f -1 (b) = a when f (a) = b AB f f -1

13 Inverse Functions Let f: A  B be a bijection, and f -1 :B  A be the inverse of f: f -1 ○ f = I A = (f -1 ○ f)(a) = f -1 (f(a)) = f -1 (b) = a f ○ f -1 = I B = (f ○ f -1 )(b) = f(f -1 (b)) = f(a) = b AB f f -1

14 Functions: Real Functions Given f :R  R and g :R  R then (f  g): R  R, is defined as (f  g)(x) = f(x)  g(x) (f. g): R  R is defined as (f g)(x) = f(x) × g(x) Example: Let f :R  R be f(x) = 2x and g :R  R be g(x) = x 3 (f+g)(x) = x 3 +2x (f. g)(x) = 2x 4