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Discrete Maths

• Objectivesto show the connection between

relations and functions, and to introduce some functions with useful, special properties

242-213, Semester 2, 2014-2015

6. Functions

1. What is a Function?A function is a special kind or relation

between two sets: f : A B

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A

f

B

domain codomain

All the elements in Ahave a single link to a value in B

Some elements in Bmay not be used. Somemay be used more thanonce.

read this as f maps A to B;it is NOT if-then

ExampleThe function f: P C with

P = {Linda, Max, Kathy, Peter}C = {Hat Yai, NY, Hong Kong, Bangkok}

Usual function notation:f(Linda) = Hat Yaif(Max) = Hat Yaif(Kathy) = Hong Kongf(Peter) = Bangkok

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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.

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the range

Range

A

f

B

2. Functions with Special PropertiesFunction ImageOne-to-one Function (injective)Onto Function (surjective)One-to-one Correspondence (bijective)Identity FunctionInverse Function

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2.1. Function Image, f(S)

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A

f

B

For some subset of A (e.g. S), the set of f() values in B are itsFunction Image f(S).

S A

f(S)

2.2. One-to-one Function

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also calledinjective

A

f

B

Each value in A maps to one value in B.

Don’t need touse all the B values.

Examples

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f(Linda) = Moscowf(Max) = Bostonf(Kathy) = Hong Kongf(Peter) = Boston

Is f one-to-one?

No, Max and Peter are mapped onto the same element of the image

g(Linda) = Moscowg(Max) = Bostong(Kathy) = Hong Kongg(Peter) = New York

Is g one-to-one?

Yes, each element is assigned a unique element of the image

2.3. Onto Function

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also calledsurjective

A

f

B

Every element in B is linked to from A.

Some elements in Bmay be used more than once.

2.4. One-to-one Correspondence

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also calledbijective

A

f

B

Each value in A maps to one value in B andevery element in B is linked to from A.

bijective == injective + surjective

Is f injective?No

Is f surjective?No

Is f bijective?No.

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Example 1

Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

f

injective: 1-1; not all Bsurjective: all Bbijective 1-1; all B

Is f injective?No

Is f surjective?Yes

Is f bijective?No

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Example 2

Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

f

Paul

injective: 1-1; not all Bsurjective: all Bbijective 1-1; all B

Is f injective?Yes

Is f surjective?No

Is f bijective?No

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Example 3

injective: 1-1; not all Bsurjective: all Bbijective 1-1; all B

Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

f

Lubeck

Is f injective?No!

f is not evena function

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Example 4

injective: 1-1; not all Bsurjective: all Bbijective 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

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Example 5

injective: 1-1; not all Bsurjective: all Bbijective 1-1; all B

Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

f

Lubeck

bijective == injective + surjective

Paul

2.5. Identity Functionf : X X is called the identity function (Ix) if every

element in X is mapped to the same elemente.g.

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12

3

4

:

Z+

12

3

4

:

Z+f(x) = x * 1f

2.6. Inverse Function

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Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

f

LubeckPaul

Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

f-1

LubeckPaul

example 5

• If f is bijective then we can create an inverse function, f-1

• if f(x) = y then f-1(y) = x

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f(Linda) = Moscowf(Max) = Bostonf(Kathy) = Hong Kongf(Peter) = Lübeckf(Paul) = New York

f-1(Moscow) = Lindaf-1(Boston) = Maxf-1(Hong Kong) = Kathyf-1(Lübeck) = Peterf-1(New York) = Paul

Inversion is only possible for bijective functions.

3. Composition of Functionsif 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

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Diagram

f g is only possible if the range of g is a subset of the domain of f

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X

Y

Z

gf

f g

f’s domaing’s range

Examplef(x) = 7x – 4, g(x) = 3x

(f g)(5) = f( g(5) ) = f(15) = 105 – 4 = 101

In general:(f g)(x) = f( g(x) ) = f(3x) = 21x - 4

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Composition and Inverse (f-1 ○ f)(x) = f-1(f(x)) = x

The composition of a function and its inverse is the identity function ix

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4. More Information

• Discrete Mathematics and its ApplicationsKenneth H. RosenMcGraw Hill, 2007, 7th edition• chapter 2, section 2.3

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