Chapter 5.4 Functions

Reading: 5.4 Next Class: 5.5

1 Motivation

 Different types of relations allow different techniques to be applied efficiently.

 Partial orderings permit to represent any ordered set or general sets with precedence constraints. – Derivation of schedules – Design of (parallel) programs

 Equivalence relations allow the representation of arbitrary partitions in sets. – Grouping of information in a database – Characterization of specific properties of elements of sets

 Functions represent an important class of relations in mathematics or for the correlation of data.

Section 4.4 Functions

1 2 Functions

 DEFINITIONS: Let S and T be sets. A function (mapping) f from S to T, f : S  T, is a subset of S  T, where each member of S appears exactly once as the first component of an ordered pair. S is the domain and T the codomain of the function. If (s, t) belongs to the function, then t is denoted by f(s); t is the image of s under f, s is a preimage of t under f, and f is said to map s to t. For A  S, f (A) denotes { f (a)  a  A}.

 f(S) is called the range of f, which is the set of all images written as R. R  T

 There are three parts to a function:

 A set of starting values

 A set from which associated values come

 The association itself Section 4.4 Functions

3 Functions (cont’d)

 The set of starting values is called the domain of the function.

 The set from which associated values come is called the codomain of the function.

 Here f is a function from S to T, symbolized f: S  T. S is the domain and T is the codomain. The association itself is a set of ordered pairs, each of the form (s,t) where s  S, t  T, and t is the value from T that the function associates with the value s from S; t = f (s).

Section 4.4 Functions

2 4 Functions (cont’d)

 A function from S to T is a subset of S  T with certain restrictions on the ordered pairs it contains.

 By the definition of a function, a binary relation that is one- to-many (or many-to-many) cannot be a function.

 Each member of S must be used as a first component.

 A function is either one-to-one or many-to-one relation.

 Note: There is always exactly ONE image for every x  S, but there can be multiple (or no) preimages for a given y  T.

 The definition of a function includes functions of more than one variable. We can have a function f : S1  S2  ...  Sn  T that associates with each ordered n-tuple of elements (s1, s2, ... , sn), si Si, a unique element of T.

Section 4.4 Functions

5 Functions (cont’d)

 The floor function x associates with each real number x the greatest integer less than or equal to x, i.e., f: R  Z, f(x) = x

 The ceiling function x associates with each real number x the smallest integer greater than or equal to x.

 2.8 = 2, 2.8 = 3. Both the floor function and the ceiling function are functions from R to Z.

 For any integer x and any positive integer n, the modulo function, denoted by f (x) = x mod n, associates with x the remainder when x is divided by n. One can write x as x = qn + r, 0  r < n, where q is the quotient and r is the remainder, so the value of x mod n is r.

 Eg. 25 mod 2 = 1 since 25 = 12 * 2 + 1, 21 mod 3 = 0 since 21 = 7*3 + 0. -17 mod 5 = 3 since -17 = (-4) * 5 + 3

Section 4.4 Functions

3 6 Functions (cont’d)

 Not all functional associations can be described by an equation. Technically, the equation only describes a way to compute associated values.

 Depending on the domain and codomain the same equation can have different properties. 3  g: R  R, where g(x) = x . 3  f : Z  R, given by f (x) = x is not the same function as g. The domain has been changed, which changes the set of ordered pairs.

Section 4.4 Functions

7 Functions (cont’d)

 DEFINITION: EQUAL FUNCTIONS Two functions are equal if they have the same domain, the same codomain, and the same association of values of the codomain with values of the domain.

 To show that two functions with the same domain and the same codomain are equal, one must show that the associations are the same.

 This can be done by showing that, given an arbitrary element of the domain, both functions produce the same associated value for that element; that is, they map it to the same place.

Section 4.4 Functions

4 8 Onto Functions

 DEFINITION: A function f: S  T is an onto, or surjective, function if the range of f equals the codomain of f.

 In every function with range R and codomain T, R  T.

 To prove that a given function is onto,

 Show that T  R; then it will be true that R = T.

 Show that an arbitrary member of the codomain is a member of the range. 3  Eg. g: R  R where g(x) = x is an onto function.

Section 4.4 Functions

9 Onto Functions (cont’d)

 For an onto, or surjective, function, every element of the codomain is the image of at least one element in the domain.

 Examples:

 f: R  R, f(x) = 2x + 3. 2  But: f: R  R, f(x) = x is not surjective.

Section 4.4 Functions

5 10 One-to-One Functions

 DEFINITION: A function f: S  T is one-to-one, or injective, if no member of T is the image under f of two distinct elements of S.

 The one-to-one idea here is the same as for binary relations in general, except that every element of S must appear as a first component in an ordered pair.

 To prove that a function is one-to-one, we assume that there are

elements s1 and s2 of S with f (s1) = f (s2) and then show that s1 = s2. 3  The function g: R  R defined by g(x) = x is one-to-one because if x and y are real numbers with g(x) = g(y), then x3 = y3 and x = y.

Section 4.4 Functions

11 One-to-One Functions (cont’d)

 For a one-to-one, or injective, function, every element in the codomain is the image of at most one element in the domain.

 Examples:

 f: N  R, f(x) = 2x + 3. 2  But: f: R  R, f(x) = x is not injective.

 But: f: {Honda, Ford, GM, Toyota} {USA, Japan}, f = {(Honda, Japan), (Ford, USA), (GM, USA), (Toyota, Japan)} is not injective.

Section 4.4 Functions

6 12 Bijections

 DEFINITION: A function f:S  T is bijective (a bijection) if it is both one-to- one and onto.

3  The function g: R  R given by g(x) = x is a bijection.

Section 4.4 Functions

13 Bijections (cont’d)

 For a bijection, every element in the domain has exactly one image in the codomain and every element of the codomain is the image of exactly one element in the domain.

 Bijections are mappings that uniquely pair one element from the domain and one element of the codomain.

 For a bijection, |S| = |T|.

Section 4.4 Functions

7 14 Composition of Functions

 If the codomain of a function f is equal to the domain of a function g, then the two functions can be composed leading to a new function. f: S  T g: T  U

g  f: S  U, (g  f )(s) = g( f (s))

 DEFINITION:

Let f: S  T and g: T  U. Then the composition function, g  f, is a function from S to U defined by (g  f )(s) = g( f (s)).

Section 4.4 Functions

15 Composition of Functions (cont’d)

 The function g  f is applied right to left; function f is applied first and then function g.

 Function composition preserves the properties of being onto and being one-to-one.

Section 4.4 Functions

8 16 Composition of Functions (cont’d)

 Examples: 2  f: R  R, f(x) = x , g: R  R, g(x) = x 2 g  f: R  R, (g  f) (x) = x  2 (g  f)(2.3) = 2.3  = 5.29 = 5 2 f  g: R  R, (f  g) (x) = ( x) 2 2 (f  g)(2.3) = ( 2.3) = 2 = 4

 Note: Order is important. Even if f  g is a valid function, g  f is often not valid. Moreover, even if both are valid functions they are often not equivalent.

 Examples: 2  f: Z  N, f(x) = x , g: N  R, g(x) = x1/2

g  f: Z  R, (g  f) (x) = |x|. f  g does not exist.

Section 4.4 Functions

17 Composition of Functions (cont’d)

 The composition of functions can be visualized using commutative diagrams. – Commutative diagrams represent the connections between domains and codomains established by a given set of functions. Composite functions can be found as “shortcuts” of a sequence of functions. f S T

g  f g

U  The composition, , of two functions is a binary operation on the set of all functions.

Section 4.4 Functions

9 18 Composition of Functions (cont’d)

 The composition of two injective functions is again an injective function.

 Proof: see the book.

 The composition of two surjective functions is again a surjective function.

 Proof: see the book.

 THEOREM ON COMPOSING TWO BIJECTIONS The composition of two bijections is a bijection.

Section 4.4 Functions

19 Inverse Functions

 Let f: S → T be a bijection. Because f is onto, every t  T has a pre-image in S. Because f is one-to-one, that pre- image is unique.

 The function that maps each element of a set S to itself, that is, that leaves each element of S unchanged, is called

the identity function on S and denoted by iS.

 DEFINITION: Let f be a function, f: S  T. If there

exists a function g: T  S such that g  f = iS and f  g = iT, then g is called the inverse function of f, denoted by f -1.

T S f f(s)=t s=g(t) g

Section 4.4 Functions

10 20 Inverse Functions (cont’d)

 A function f has an inverse if and only if f is a bijection.

 Examples:

 f: z  z, f(x) = x - 5, f -1: z  z, f -1 (x) = x + 5.

 THEOREM ON BIJECTIONS AND INVERSE FUNCTIONS Let f: S → T. Then f is a bijection if and only if f -1 exists.

Section 4.4 Functions

21 Exercise

 Let S = {1, 2, 3, 4}, T = {1, 2, 3, 4, 5, 6}, and U = {6, 7, 8, 9, 10}. Also let f = {(1, 2), (2, 4), (3, 3), (4, 6)} be a function from S to T, and let g = {(1, 7), (2, 6), (3, 9), (4, 7), (5, 8), (6, 9)} be a function from T to U.

Write the ordered pairs in the function g  f .

Section 4.4 Functions

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