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Section 1.8

Functions

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Loose Definition

• Mapping of each element of one set onto some element of another set– each element of 1st set must map to something,

but that something need not be unique; 2 or more elements of 1st set can map to single element in 2nd set

– however, no element of 1st set can map to more than one element of 2nd set

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Examples

Let A = {x, y, z} and B = {1, 2, 3}

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Formal Definition

• Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A

• In more algebraic terms:

f(a) = b if b B and is the unique element assigned to a A

• If f is a function from A to B we write

f: A B

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Specifying Functions

• Can explicitly state assignments, e.g. f(x)=2, f(y)=1, f(z)=3

• Can write as formula, e.g. f(x)=x+1

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Some Terminology

• Given f:AB– A is the domain of f– B is the co-domain of f

• Given f(a)=b– b is the image of a– a is the pre-image of b

• The range of f is the set of all images of elements of A

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Examples

f(x)=2, f(y)=1, f(z)=3domain is {x,y,z}co-domain is {1,2,3}range is {1,2,3}

f(x)=2, f(y)=2, f(z)=2domain is {x,y,z}co-domain is {1,2,3}range is {2}

Suppose A=N and B=N and f:AB = f(x) = x*2Then the domain and co-domain are N; the range is thepositive even integers

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Addition & Multiplication of Functions

Two real-valued functions with the same domain can be addedand multiplied

Where f1: AR and f2: A R, (f1 + f2)(x) = f1(x) + f2(x) and (f1f2)(x) = f1(x) * f2(x)

For example, let f1: RR = f1(x) = x + 2 and f2: RR = f2(x) = x2 + 3

(f1 + f2)(x) = f1(x) + f2(x) = (x + 2) + (x2 + 3) = x2 + x + 5 (f1f2)(x) = f1(x) * f2(x) = (x + 2)(x2 + 3) = x3 + 2x2 + 3x + 6

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Image of a SubsetGiven f:AB and SAThe image of S is the subset of B that consists of the imagesof the elements of S:

f(S) = {f(s) | s S}For example:

Suppose S = {x,y}Then the image of S is the setf(S) = {3,1}

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One-to-one, or Injective Functions

• If each member of set A has a unique image in function f, then the domain of f:AB is said to be a one-to-one function

• A one-to-one function is also called an injection

• A function is injective if and only if f(x) = f(y) implies that x=y in the domain of f

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ExamplesLet A = Z and B = Z and f:AB = f(n) = n - 1

Suppose n = x = yIf x = y then x-1 = y - 1

So f is one-to-one

Let A = Z and B = Z and f:AB = f(n) = n2 + 1Suppose n = x = yIf x = y then x2 + 1 = y2 + 1, and x2 = y2

But, for example, -22 = 22 So f is not one-to-one

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Strictly Increasing/Decreasing Functions

• If AR and BR and f:AB and x & y are in the domain of f,

• If f(x) < f(y) whenever x<y, then f is said to be strictly increasing

• If f(x) > f(y) whenever x<y, then f is said to be strictly decreasing

• All such functions are one-to-one

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Surjective (Onto) Functions

• A function f:AB is surjective if and only if for every element b B, there is an element a A with f(a) = b

• In other words, if all elements in B have an A element or elements mapped to them, it’s a surjective function

• Or, all elements in co-domain are images of elements in domain; range = co-domain

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Bijection: One-to-one Correspondence

• If a function is BOTH injective and surjective (one-to-one and onto), it is bijective

• If A is a finite set, and f is a function from A to itself (f:AA), then f is injective ONLY if it is surjective

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Identity function on a set

The identity function assigns each elementof a set to itself

iA: AA where iA(x) = x where x A

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Inverse Function

• Given f:AB, and f is a bijection

• The inverse function of f, denoted f -1, assigns to an element bB the unique element a A such that f(a)=b

• In other words, when f(a)=b, f -1(b)=a

• A bijection is invertible because its inverse can be defined; a function that is not a bijection is not invertible

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Composition of 2 functions

• Given two functions, f and g such that g:AB and f:BC,

• The composition of f and g, denoted (f o g)(a), is f(g(a))

• Take the result of g(a) and plug it into f to get (f o g)(a)

• f o g can only be defined if the range of g is a subset of the range of f

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Example

Find f o g and g o f where f:RR = f(x) = x2 + 1 and g:RR = g(x) = x + 2

f o g = f(g(x)) = f(x+2) = (x+2)2 + 1 = x2 + 4x + 5

g o f = g(f(x)) = g(x2 + 1) = (x2 + 1) + 2 = x2 + 3

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Notes on Composition

• As is evident from the previous example, the commutative law does not apply to composition; in other words, f o g g o f

• When the composition of a function and its inverse is found, an identity function is obtained: (f -1)-1 = f

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Graphs of Functions

• The graph of a function is a set of ordered pairs

• For f:AB, the graph of f is the set defined as: { a,b | a A and b B }

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Floor & Ceiling Functions

• Floor function: assigns to real number x the largest whole number that is less than or equal to x - denoted x or [x]

• Ceiling function: assigns to real number x the smallest whole number that is greater than or equal to x - denoted x

• These functions have useful applications involving the storage & transmission of data

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ExampleHow many bytes are required to encode 11,325bits of data for transmission (as strings of 8-bitbytes)?Dividing 11,325 bits by 8 bits per byte producesthe result 1415.625

Since we can’t transmit anything smaller than a byte, we use the ceiling function to find theclosest usable whole number: 1415.625 = 1416

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Section 1.6

Functions

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