Discrete Maths Objectives to show the connection between relations and functions, and to introduce some functions with useful, special properties , Semester 2, Functions 1

1. What is a Function? A function is a special kind or relation between two sets: f : A  B 2 A f B domaincodomain All the elements in A have a single link to a value in B All the elements in A have a single link to a value in B Some elements in B may not be used. Some may be used more than once. Some elements in B may not be used. Some may be used more than once. read this as f maps A to B; it is NOT if-then read this as f maps A to B; it is NOT if-then

Example The function f: P  C with P = {Linda, Max, Kathy, Peter} C = {Hat Yai, NY, Hong Kong, Bangkok} Usual function notation: f(Linda) = Hat Yai f(Max) = Hat Yai f(Kathy) = Hong Kong f(Peter) = Bangkok 3 Linda Max Kathy Peter Bangkok Hong Kong NY Hat Yai P C f

The range is the set of elements in B used by the function f. 4 the range Range A f B

2. Functions with Special Properties Function Image One-to-one Function (injective) Onto Function (surjective) One-to-one Correspondence (bijective) Identity Function Inverse Function 5

2.1. Function Image, f(S) 6 A f B For some subset of A (e.g. S), the set of f() values in B are its Function Image f(S). For some subset of A (e.g. S), the set of f() values in B are its Function Image f(S). S  A f(S)

2.2. One-to-one Function 7 also called injective also called injective A f B Each value in A maps to one value in B. Don’t need to use all the B values.

Examples 8 f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Boston Is f one-to-one? No, Max and Peter are mapped onto the same element of the image g(Linda) = Moscow g(Max) = Boston g(Kathy) = Hong Kong g(Peter) = New York Is g one-to-one? Yes, each element is assigned a unique element of the image

2.3. Onto Function 9 also called surjective also called surjective A f B Every element in B is linked to from A. Some elements in B may be used more than once.

2.4. One-to-one Correspondence 10 also called bijective also called bijective A f B Each value in A maps to one value in B and every element in B is linked to from A. Each value in A maps to one value in B and every element in B is linked to from A. bijective == injective + surjective

Is f injective? No Is f surjective? No Is f bijective? No. 11 Example 1 Linda Max Kathy Peter Boston New York Hong Kong Moscow f injective: 1-1; not all B surjective: all B bijective 1-1; all B injective: 1-1; not all B surjective: all B bijective 1-1; all B

Is f injective? No Is f surjective? Yes Is f bijective? No 12 Example 2 Linda Max Kathy Peter Boston New York Hong Kong Moscow f Paul injective: 1-1; not all B surjective: all B bijective 1-1; all B injective: 1-1; not all B surjective: all B bijective 1-1; all B

Is f injective? Yes Is f surjective? No Is f bijective? No 13 Example 3 injective: 1-1; not all B surjective: all B bijective 1-1; all B injective: 1-1; not all B surjective: all B bijective 1-1; all B Linda Max Kathy Peter Boston New York Hong Kong Moscow f Lubeck

Is f injective? No! f is not even a function 14 Example 4 injective: 1-1; not all B surjective: all B bijective 1-1; all B injective: 1-1; not all B surjective: all B bijective 1-1; all B Linda Max Kathy Peter Boston New York Hong Kong Moscow f Lubeck

Is f injective? Yes Is f surjective? Yes Is f bijective? Yes 15 Example 5 injective: 1-1; not all B surjective: all B bijective 1-1; all B injective: 1-1; not all B surjective: all B bijective 1-1; all B Linda Max Kathy Peter Boston New York Hong Kong Moscow f Lubeck bijective == injective + surjective Paul

2.5. Identity Function f : X  X is called the identity function (I x ) if every element in X is mapped to the same element e.g : Z : f(x) = x * 1 f

2.6. Inverse Function 17 Linda Max Kathy Peter Boston New York Hong Kong Moscow f Lubeck Paul Linda Max Kathy Peter Boston New York Hong Kong Moscow f -1 Lubeck Paul example 5 If f is bijective then we can create an inverse function, f -1 if f(x) = y then f -1 (y) = x

18 f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Lübeck f(Paul) = New York f -1 (Moscow) = Linda f -1 (Boston) = Max f -1 (Hong Kong) = Kathy f -1 (Lübeck) = Peter f -1 (New York) = Paul Inversion is only possible for bijective functions.

3. Composition of Functions if g: X  Y and f: Y  Z then f  g: X  Z read  as “first do g then do f” The composition of f and g is defined as: (f  g)(x) = f( g(x) ) this is why f and g are ordered this way 19

Diagram f  g is only possible if the range of g is a subset of the domain of f 20 X Y Z g f f  g f’s domain g’s range

Example f(x) = 7  x – 4, g(x) = 3  x (f  g)(5) = f( g(5) ) = f(15) = 105 – 4 = 101 In general: (f  g)(x) = f( g(x) ) = f(3  x) = 21  x

Composition and Inverse (f -1 ○ f)(x) = f -1 (f(x)) = x The composition of a function and its inverse is the identity function i x 22

4. More Information Discrete Mathematics and its Applications Kenneth H. Rosen McGraw Hill, 2007, 7th edition chapter 2, section