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1
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
3liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
4liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Assignment of grades in Discrete Mathematic Class
FUNCTIONS
5liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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?
6liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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?
7liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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?
8liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
f(x) = {(1,2), (2,3), (3,4)}FuntionExactly one element in y is associated with
every element in x
IS THIS A FUNCTION?
9liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Everywhere definedDomainCodomainRange ImagePre-image/Inverse image
TERMS IN FUNCTON
10liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
11liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
12liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
13liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
One to OneOnto CompositionInverse
PROPERTIES OF FUNCTION
14liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
15liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
All element in the domain is assigned to element in domain
All elements in codomain has assignmentAlso known as surjective.
ONTO FUNCTION
16liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
17liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
18liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
19liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Find .
Let
INVERSE
20liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
(2)
Let
=
= 1.2247
INVERSE
21liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Let
INVERSE
22liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Find inverse of Find of functionFunction Find the value of when
EXERCISES
23liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
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
24liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
COMPOSITE
25liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Given and Find each of the followings:
EXERCISES
26liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Find
Assume
COMPOSITE
27liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Find Solve the equation
EXERCISE
28liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Two important function in discrete Mathematics: Floor Function Ceiling Function
GRAPH FUNCTION
29liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
The floor function rounds x down to the closest integer less than or equal to x.
Example :
FLOOR FUNCTION
30liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Draw a graph for
FLOOR FUNCTION
31liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
The ceiling function round x up to the closest integer greater than or equal to x
Example :
CEILING FUNCTION
32liyana JMSK , POLITEKNIK BALIK PULAU
BA202 Discrete Mathematics
Draw a graph for
CEILING FUNCTION