Chapter 7 Functions

Chapter 7 Functions Dr. Curry Guinn

Outline of Today • Section 7.1: Functions Defined on General Sets • Section 7.2: One-to-One and Onto • Section 7.3: The Pigeonhole Principle • Section 7.4: Composition of functions • Section 7.5: Cardinality

What is a function? • A function f f : X → Y • Maps a set X to a set Y • is a relation between • the elements of X (called the inputs) and • the elements of Y (called the outputs) • with the property that each input is related to one and only one output. • X is the domain. • Y is the co-domain. • The set of all values f(x) is called the range.

How do we represent functions • Arrow diagrams • f : X → Y • What is the domain? • {a,b,c,d} • What is the co-domain? • {x, y, z, p, q, r, s} • What is the range? • {x, y, p} • What is the inverse image of y? • {a, c}

How do we represent functions? • As Ordered Pairs • f = { (a,y), (b,p), (c,y), (d,x) }

How do we represent functions? • As machines

How do we represent functions? • By Formula • f(x) = 2x2 + 3

Equality of functions • Two functions, f and g, are equal if • Both map from set X to set Y • And • f(x) = g(x) for all x  X. • If f(x) = SQRT(x^2) and g(x) = x, is f = g? • Identity function, i is such that • i(x) = x for all x  X

The Logarithmic Function • Logb x = y  by = x • Log2 8 • Log10 1000 • Log3 3n • Log5 1/25 • Loga 1 for a > 0

“Well defined” function • Remember: a function must map an input to a single, unique value • F: R → R such that • f(x) = SQRT(-x2) for all real numbers X. • Why is this not well defined?

7.2: One-to-one and onto • A function is f : X → Y is one-to-one when • If f(x1) = f(x2), then x1 must be equal to x2. • To show a function is one-to-one, assume f(a) = f(b) for arbitrary a and b. Show a = b.

One-to-one and finite sets • See board

One-to-one example, infinite sets • f(n) = 2n + 1 • f(n) = n^2

Onto functions • A function f : X → Y is onto if for every y in the co-domain, that is, every y  Y, there exist some x  X, such that f(x) = y. • To prove something is onto, pick an arbitrary element in Y and find an x in X that maps to the y.

Onto functions and finite sets • See board

Onto examples, infinite sets • Show the following are or are not onto: • f: Z → Z by f(n) = 2n + 1. • f: Z → Z by f(n) = n + 5.

One-to-one correspondences • If a function f : X → Y is both one-to-one and onto, there is a one-to-one correspondence (or bijection) from the set X to set Y. • Show f: Z → Z by f(n) = n + 5 has a one-to-one correspondence.

Inverse functions • If a function f : X → Y is has as one-to-one correspondence, the there is an inverse function f-1: Y → X such that if f(x) = y, then f-1(y) = x. • f(x) = n + 5 • What’s the inverse?

Exponential and Logarithmic functions • Expb(x) = bx for any x  R and b > 0. • Logb(x) = y, for any x  R+ if x = by • Show logb(x/y) = logb(x) – logb(y) • Hint: Let u = logb(x) and v = logb(y)

7.3: The Pigeonhole Principle • Suppose X and Y are finite sets and N(X) > N(Y). Then a function f : X → Y cannot be one-to-one. • Proof by contradiction.

Using the Pigeonhole Principle • Prove there must be at least 2 people in New York city with the same number of hairs on their head. • How many integers must you pick in order for them to have at least one pair with the same remainder when divided by 3.

The Generalized Pigeon Hole Principle • For any function f from a finite set X to a finite set Y and for any positive integer k, if N(X) > k*N(Y), then there is some y  Y such that the inverse image of y has at least k+1 distinct elements of X. • If you have 85 people, and there are 26 possible initials of their last name, at least one initial must be used at least ___ times.

Pigeonhole • In a group of 1,500 people, must at least five people have the same birthday?

7.4 Composition of Functions

Composition of functions • The composition of two functions occurs when the output of one function is the input to another. • Let f: X → Y’ and g: Y → Z where the range of f is a subset of the domain of g. Define a new function g  f(x) = g(f(x)).

Composition Examples • F(n) = n + 1 and g(n) = n2 • What is f  g? • What is g  f?

Composition of one-to-one functions • If both f and g are one-to-one, is f  g one-to-one? • Proof: See board • If both f and g are onto, is f  g onto?

7.5 Cardinality • The cardinality of a set is how many members it has. • Let X and Y be sets. X has the same cardinality as Y iff there exists a one-to-one correspondence from X to Y. • X has the same cardinality as X (Reflexive) • If X has the same cardinality as Y, Y has the same cardinality as X (Symmetric) • If X has the same cardinality as Y, and Y has the same cardinality as Z, then x has the same cardinality as Z (Transitive).

Countable Sets • A set X is countably infinite iff has the same cardinality as the set of positive integers. • Is the set of all integers countable? • F(n) = { 0 if n = 1 -n/2 if n is even (n-1)/2 if n is odd } Is this one-to-one? Onto?

Rational numbers are countable

The set of real numbers between 0 and 1 is uncountable. • Uses Cantor’s Diagnolization Argument • Proof by contradiction • See board!! • Any set with an uncountable subset is uncountable.

Some interesting results • The set of all computer programs in a given computer language is countable. • How? • Each program is a finite set of strings. • Convert to binary. • Now each program is a unique number in the set of integers. • The set of all programs is a subset of the set of all integers. Therefore countable.

Chapter 7 Functions