ba202 ch2.3 function

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2.3 EXPLAIN FUNCTION

CLO3: SOLVE RELATED PROBLEMS CRITICALLY USING APPROPRIATE FORMULAE AND CONCEPTS. (C3, A1)

2liyana JMSK , POLITEKNIK BALIK PULAU

BA202 Discrete Mathematics

Describe basic constructions.Explain the properties of following functions:

One-to-one functions Onto functions Composition functions Inverse functions

Describe graphs of the Floor and Ceiling functions.

LEARNING OUTCOMES



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

Write as (Function from A to B)Function sometimes called mapping.

(Mapping input to output)

FUNCTIONS



Assignment of grades in Discrete Mathematic Class

FUNCTIONS



R = { (1,a), (2,b), (3,b), (4, c)R is a function from A to B. Exactly one

element in B is assigned to every element of A.

IS THIS A FUNCTION?



R= {(1,a), (2,b), 3,c)}R is not a functionBecause 4 belongs to A and 4 is not

associated with any element of B.R is only a relation but not a function.

IS THIS A FUNCTION?



R= {(1,a), (1,b), (2,c), (3,c), (4,c)Not functionBecause 1 belongs to A and 1 is associated

with two element of B.

IS THIS A FUNCTION?



f(x) = {(1,2), (2,3), (3,4)}FuntionExactly one element in y is associated with

every element in x

IS THIS A FUNCTION?



Everywhere definedDomainCodomainRange ImagePre-image/Inverse image

TERMS IN FUNCTON



Let f be a function from A to B. Then we say that f is everywhere defined function if Dom(f) = A.

In other words, all domain are used.

EVERYWHERE DEFINED

Everywhere defined Not everywhere defined



f is a function from A to B represented by the diagram above:

Domain : All element in A { Adam, Chou, Ali, Steven, Jacob }

Codomain : All element in B Codomain : { A, B, C, D, E }

Range : Element in B that associated with element in A { A, B, C, E }

TERMS IN FUNCTION



Image : If f(a) = b, the image of a is bf(Adam) = A

The image of Adam is A The image of Jacob is E

Pre-image or inverse image Pre-image of A are Adam and Steven Pre-image of B is Ali

Tambah objek

TERMS IN FUNCTION



One to OneOnto CompositionInverse

PROPERTIES OF FUNCTION



Function that assign one value in the domain to one value in codomain and

Never assign the same value of codomain to two different domain elements.

Also known as injective.

ONE TO ONE FUNCTION



All element in the domain is assigned to element in domain

All elements in codomain has assignmentAlso known as surjective.

ONTO FUNCTION



A function is one to one correspondence if the function satisfy the properties below: one to one and onto

Also known as bijection.

ONE TO ONE CORRESPONDENCE



To define Inverse Function, the function has to be one to one correspondence.

A function is not invertible if it is not a one-to-one correspondence, because the inverse of such function does not exist.

INVERSE

inverse

• It is not a function c is element in the domain but not associated with any element in codomain

• element a is assigned to two different element in codomain



To obtain the inverse function, simply reversing the direction of each arrow.

For example : = {(1,a), (2,c), (3,b)} = { (a, 1), (c,2), (b,3) }

INVERSE



Find .

Let

INVERSE



(2)

Let

=

= 1.2247

INVERSE



Let

INVERSE



Find inverse of Find of functionFunction Find the value of when

EXERCISES



Combination of two functions.If we have two functions, f(x) and g(x) the

composite of f(x) and g(x) can be denote as : fg(x) f ○ g

COMPOSITE



COMPOSITE



Given and Find each of the followings:

EXERCISES



Find

Assume

COMPOSITE



Find Solve the equation

EXERCISE



Two important function in discrete Mathematics: Floor Function Ceiling Function

GRAPH FUNCTION



The floor function rounds x down to the closest integer less than or equal to x.

Example :

FLOOR FUNCTION



Draw a graph for

FLOOR FUNCTION



The ceiling function round x up to the closest integer greater than or equal to x

Example :

CEILING FUNCTION



Draw a graph for

CEILING FUNCTION

ba202 ch2.3 function

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