Understanding Functions: Definitions, Properties, and Types in Discrete Mathematics

1. 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 imageof 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. John Smith Edward Jones Richard Boone Mike Mario Kim Joe Jill Example f A B Domain Codomain • 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) = x2 • -- 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. f g 2     h 3 b 5 d 1 o 7 A B C Function Composition Given the functions g:ABand f:BC, the compositionof f and g, f ○g: AC defined as f○g (a) = f ( g (a) ) f○g (h) ?

7. Function Composition Properties • Associative: Given the functions g:ABand f:BC and h:CDthen 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. f A B 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

10. f A B 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

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

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 A B 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 = IA = (f-1○f)(a) = f-1 (f(a)) = f-1 (b) = a f ○ f-1 = IB = (f○f-1)(b) = f(f-1 (b)) = f(a) = b A B 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) = x3 (f+g)(x) = x3+2x (f . g)(x) = 2x4