Discrete Mathematics and Its Applicationsdase.ecnu.edu.cn/mgao/teaching/DM_2020_Spring/slides/2_sets23.pdf · Let A and B be nonempty sets. A function f from A to B is an assignment

Discrete Mathematics and Its ApplicationsLecture 2: Basic Structures: Function

MING GAO

DaSE@ ECNU(for course related communications)

[email protected]

Mar. 27, 2020

Outline

1 Function

2 One-to-one and onto functions

3 Inverse Functions and Compositions of Functions

4 Some important functions

5 Take-aways

MING GAO (DaSE@ECNU) Discrete Mathematics and Its Applications Mar. 27, 2020 2 / 25

Function

Motivations

Classifier

Moore’s law Complexity analysis

Motivations

Functions connect the relationships between different objects;

A function may tell us a rule;

Models in data mining, machine learning, etc., are usuallypresented in functions.


Function

Motivations

Classifier Moore’s law

Complexity analysis

Motivations





Function

Motivations

Classifier Moore’s law Complexity analysis

Motivations





Function

Function

Definition

Let A and B be nonempty sets. A function f from A to B is anassignment of exactly one element of B to each element of A. Wewrite f (a) = b if b is the unique element of B assigned by function fto element a of A. If f is a function from A to B, we write f : A→ B.

Functions are sometimes also called mappings or transformations;

Functions are specified in many different ways, such as assignmentrules, formula, relations, etc;

This function is defined by assignment f (a) = b, where (a, b) is theunique ordered pair in the relation that has a as its first element.


Function

Function

Definition






Function

Function

Definition






Function

Function

Definition






Function

Function

Definition






Function

Domain and codomain

Definition

If f is a function from A to B, we say that A is the domain of f andB is the codomain of f . If f (a) = b, we say that b is the image ofa and a is a preimage of b. The range, or image, of f is the set ofall images of elements of A. Also, if f is a function from A to B, wesay that f maps A to B.

Let g be the function that assigns a grade to a student in ourdiscrete mathematics class. Note that g(Adams) = A, forinstance.

The domain of g is set{Adams,Chou,Goodfriend ,Rodriguez ,Stevens}, and thecodomain is set {A,B,C ,D,F}.The range of g is set {A,B,C ,F}, because each grade exceptD is assigned to some students.


Function

Domain and codomain

Definition





Function

Domain and codomain

Definition



The domain of g is set{Adams,Chou,Goodfriend ,Rodriguez ,Stevens}, and thecodomain is set {A,B,C ,D,F}.

The range of g is set {A,B,C ,F}, because each grade exceptD is assigned to some students.


Function

Domain and codomain

Definition





Function

Add and product

Definition

Let f1 and f2 be functions from A to R. Then f1 + f2 and f1f2 arealso functions from A to R defined for all x ∈ A by

(f1 + f2)(x) = f1(x) + f2(x),

(f1f2)(x) = f1(x)f2(x).

Example

Let f1 and f2 be functions from R to R such that f1(x) = (x + 1)2

and f2(x) = −(x − 1)2. Thus, we have

(f1 + f2)(x) = f1(x) + f2(x) = (x + 1)2 − (x − 1)2 = 4x ,

(f1f2)(x) = f1(x)f2(x) = −(x + 1)2(x − 1)2 = −(x2 − 1)2.


Function

Add and product

Definition

Let f1 and f2 be functions from A to R. Then f1 + f2 and f1f2 arealso functions from A to R defined for all x ∈ A by

(f1 + f2)(x) = f1(x) + f2(x),

(f1f2)(x) = f1(x)f2(x).

Example

Let f1 and f2 be functions from R to R such that f1(x) = (x + 1)2

and f2(x) = −(x − 1)2. Thus, we have

(f1 + f2)(x) = f1(x) + f2(x) = (x + 1)2 − (x − 1)2 = 4x ,

(f1f2)(x) = f1(x)f2(x) = −(x + 1)2(x − 1)2 = −(x2 − 1)2.


Function

Projection

Definition

Let f be a function from A to B and let S be a subset of A. Theimage of S under f is the subset of B that consists of the images ofthe elements of S . We denote the image of S by f (S), so

f (S) = {t|∀s ∈ S , t = f (s)}.

Example

Let f (x) be function from R to R such that f (x) = x + 1, and Z bethe set of integers. Then

f (Z ) = Z .


Function

Projection

Definition

Let f be a function from A to B and let S be a subset of A. Theimage of S under f is the subset of B that consists of the images ofthe elements of S . We denote the image of S by f (S), so

f (S) = {t|∀s ∈ S , t = f (s)}.

Example

Let f (x) be function from R to R such that f (x) = x + 1, and Z bethe set of integers. Then

f (Z ) = Z .


One-to-one and onto functions

One-to-one function

Definition

A function f is said to be one-to-one, or an injunction, if and only iff (a) = f (b) implies that a = b for all a and b in the domain of f . Afunction is said to be injective if it is one-to-one.

Injunctive function

Remark: f is one-to-one ⇔ ∀a∀b(f (a) = f (b) → a = b) or∀a∀b(a 6= b → f (a) 6= f (b)).



One-to-one function

Definition

A function f is said to be one-to-one, or an injunction, if and only iff (a) = f (b) implies that a = b for all a and b in the domain of f . Afunction is said to be injective if it is one-to-one.

Injunctive function

Remark: f is one-to-one ⇔ ∀a∀b(f (a) = f (b) → a = b) or∀a∀b(a 6= b → f (a) 6= f (b)).



Onto function

Definition

A function f from A to B is called onto, or a surjection, if and onlyif for every element b ∈ B there is an element a ∈ A with f (a) = b.A function f is called surjective if it is onto.

Surjective function

Remark: A function f is onto if ∀y∃x(f (x) = y).



Onto function

Definition

A function f from A to B is called onto, or a surjection, if and onlyif for every element b ∈ B there is an element a ∈ A with f (a) = b.A function f is called surjective if it is onto.

Surjective function

Remark: A function f is onto if ∀y∃x(f (x) = y).



Monotonicity

Definition

A function f whose domain and codomain are subsets of the set ofreal numbers is called increasing if f (x) ≤ f (y), and strictly increas-ing if f (x) < f (y), whenever x < y and x and y are in the domainof f . Similarly, f is called decreasing if f (x) ≥ f (y), and strictlydecreasing if f (x) > f (y), whenever x < y and x and y are in thedomain of f .

Remarks

A function f is increasing if ∀x∀y(x < y → f (x) ≤ f (y)),strictly increasing if ∀x∀y(x < y → f (x) < f (y));

A function f is decreasing if ∀x∀y(x < y → f (x) ≥ f (y)),strictly increasing if ∀x∀y(x < y → f (x) > f (y)).



Monotonicity

Definition

A function f whose domain and codomain are subsets of the set ofreal numbers is called increasing if f (x) ≤ f (y), and strictly increas-ing if f (x) < f (y), whenever x < y and x and y are in the domainof f . Similarly, f is called decreasing if f (x) ≥ f (y), and strictlydecreasing if f (x) > f (y), whenever x < y and x and y are in thedomain of f .

Remarks

A function f is increasing if ∀x∀y(x < y → f (x) ≤ f (y)),strictly increasing if ∀x∀y(x < y → f (x) < f (y));

A function f is decreasing if ∀x∀y(x < y → f (x) ≥ f (y)),strictly increasing if ∀x∀y(x < y → f (x) > f (y)).


Inverse Functions and Compositions of Functions

One-to-one correspondence

Definition

Function f is a one-to-one correspondence, or a bijection, if it is bothone-to-one and onto. We also say that such a function is bijective.

Remarks

Remarks: f is a one-to-one correspondence if and only if ⇔∀a∀b(a = b ↔ f (a) = f (b)).



One-to-one correspondence

Definition

Function f is a one-to-one correspondence, or a bijection, if it is bothone-to-one and onto. We also say that such a function is bijective.

Remarks

Remarks: f is a one-to-one correspondence if and only if ⇔∀a∀b(a = b ↔ f (a) = f (b)).



Inverse function

Definition

Let f be a one-to-one correspondence from set A to set B. Theinverse function of f is the function that assigns an element b ∈ Bto a unique element a ∈ A such that f (a) = b. The inverse functionof f is denoted by f −1. Hence, f −1(b) = a when f (a) = b.

Remarks

Remarks:

f −1 6= 1f ;

A one-to-onecorrespondence is calledinvertible because we candefine an inverse of thisfunction.



Inverse function

Definition


Remarks

Remarks:

f −1 6= 1f ;




Inverse function

Definition


Remarks

Remarks:

f −1 6= 1f ;




Composition of functions

Definition

Let g be a function from set A to set B and let f be a function fromset B to set C . The composition of functions f and g , denoted forall a ∈ A by f ◦ g , is defined by

(f ◦ g)(a) = f (g(a)).

Remarks

Remarks: The com-mutative law does nothold for the composi-tion of functions, i.e.,

f ◦ g 6= g ◦ f .




Definition


(f ◦ g)(a) = f (g(a)).

Remarks


f ◦ g 6= g ◦ f .




Definition


(f ◦ g)(a) = f (g(a)).

Remarks


f ◦ g 6= g ◦ f .



Examples

Example I

Let f (x) = 2x + 3 and g(x) = 3x + 2 from Z to Z . What is thecomposition of f and g? What is the composition of g and f ?

(f ◦ g)(x) = f (g(x)) = f (3x + 2) = 2(3x + 2) + 3 = 6x + 7;

(g ◦ f )(x)

Example II

If f (x) = 3x + 2 and g(x) = 13x −

23 , how about the answers?

(f ◦ g)(x) = f (g(x)) = f (13x −23) = 3(13x −

23) + 2 = x ;

(g ◦ f )(x) = g(f (x)) = g(3x + 2) = 13(3x + 2)− 2

3 = x .

Remarks:

(f ◦ f −1)(b) = f (f −1(b)) = f (a) = b, i.e., (f ◦ f −1) = IB ;

(f −1 ◦ f )(a) = f −1(f (a)) = f −1(b) = a, i.e., (f −1 ◦ f ) = IA.



Examples

Example I

Let f (x) = 2x + 3 and g(x) = 3x + 2 from Z to Z . What is thecomposition of f and g? What is the composition of g and f ?

(f ◦ g)(x) = f (g(x)) = f (3x + 2) = 2(3x + 2) + 3 = 6x + 7;

(g ◦ f )(x)

Example II

If f (x) = 3x + 2 and g(x) = 13x −

23 , how about the answers?

(f ◦ g)(x) = f (g(x)) = f (13x −23) = 3(13x −

23) + 2 = x ;

(g ◦ f )(x) = g(f (x)) = g(3x + 2) = 13(3x + 2)− 2

3 = x .

Remarks:

(f ◦ f −1)(b) = f (f −1(b)) = f (a) = b, i.e., (f ◦ f −1) = IB ;

(f −1 ◦ f )(a) = f −1(f (a)) = f −1(b) = a, i.e., (f −1 ◦ f ) = IA.



Inverse of composition of functions

Theorem

Let f and g be invertible functions such that their composition f ◦ gis well defined. Then f ◦ g is invertible and (f ◦ g)−1 = g−1 ◦ f −1.

Proof.

Let A, B, and C be sets such that g : A→ B and f : B → C .Then the following two equations must be shown to hold:(g−1 ◦ f −1) ◦ (f ◦ g) = IA, and (f ◦ g) ◦ (g−1 ◦ f −1) = IB .

In terms of associative rule for function composition (homework),we have

(g−1 ◦ f −1) ◦ (f ◦ g) = g−1 ◦ ((f −1 ◦ f ) ◦ g)

= g−1 ◦ (IB ◦ g)

= g−1 ◦ g = IA

Similarly, we have (f ◦ g) ◦ (g−1 ◦ f −1) = IB .



Inverse of composition of functions

Theorem

Let f and g be invertible functions such that their composition f ◦ gis well defined. Then f ◦ g is invertible and (f ◦ g)−1 = g−1 ◦ f −1.

Proof.

Let A, B, and C be sets such that g : A→ B and f : B → C .Then the following two equations must be shown to hold:(g−1 ◦ f −1) ◦ (f ◦ g) = IA, and (f ◦ g) ◦ (g−1 ◦ f −1) = IB .In terms of associative rule for function composition (homework),we have

(g−1 ◦ f −1) ◦ (f ◦ g) = g−1 ◦ ((f −1 ◦ f ) ◦ g)

= g−1 ◦ (IB ◦ g)

= g−1 ◦ g = IA

Similarly, we have (f ◦ g) ◦ (g−1 ◦ f −1) = IB .MING GAO (DaSE@ECNU) Discrete Mathematics and Its Applications Mar. 27, 2020 15 / 25


Graph of functions

Definition

Let f be a function from set A to set B. The graph of function f isthe set of ordered pairs

{(a, b)|a ∈ A ∧ f (a) = b}.

Examples

f (n) = 2n + 1 from Z to Z f (x) = x2 + 1 from Z to Z



Graph of functions

Definition

Let f be a function from set A to set B. The graph of function f isthe set of ordered pairs

{(a, b)|a ∈ A ∧ f (a) = b}.

Examples

f (n) = 2n + 1 from Z to Z f (x) = x2 + 1 from Z to ZMING GAO (DaSE@ECNU) Discrete Mathematics and Its Applications Mar. 27, 2020 16 / 25

Some important functions

Some important functions——Floor and ceiling functions

Definition

The floor function assigns to a real number x the largestinteger that is less than or equal to x . The value of the floorfunction at x is denoted by bxc;The ceiling function assigns to a real number x the smallestinteger that is greater than or equal to x . The value of theceiling function at x is denoted by dxe.

Remarks

bxc = max {m ∈ Z : m ≤ x};dxe = min {m ∈ Z : m ≥ x}.There are many applications, including data transmission, anddata storage, etc;



Some important functions——Floor and ceiling functions

Definition

The floor function assigns to a real number x the largestinteger that is less than or equal to x . The value of the floorfunction at x is denoted by bxc;The ceiling function assigns to a real number x the smallestinteger that is greater than or equal to x . The value of theceiling function at x is denoted by dxe.

Remarks

bxc = max {m ∈ Z : m ≤ x};dxe = min {m ∈ Z : m ≥ x}.There are many applications, including data transmission, anddata storage, etc;



Graphs of floor and ceiling functions

Graphs

Floor function Ceiling function

Examples

b12c = 0, d12e = 1;

b−12c = −1, d−1

2e = 0;



Graphs of floor and ceiling functions

Graphs

Floor function Ceiling function

Examples

b12c = 0, d12e = 1;

b−12c = −1, d−1

2e = 0;



Useful properties of floor and ceiling functions

Properties

Let n be an integer, x be a real number.

bxc = n if and only if n ≤ x < n + 1;

dxe = n if and only if n − 1 < x ≤ n;

bxc = n if and only if x − 1 < n ≤ x ;

dxe = n if and only if x ≤ n < x + 1;

x − 1 < bxc ≤ x ≤ dxe < x + 1;

b−xc = −dxe;d−xe = −bxc;bx + nc = bxc+ n;

dx + ne = dxe+ n




Properties






x − 1 < bxc ≤ x ≤ dxe < x + 1;


dx + ne = dxe+ n




Properties






x − 1 < bxc ≤ x ≤ dxe < x + 1;


dx + ne = dxe+ n




Properties






x − 1 < bxc ≤ x ≤ dxe < x + 1;


dx + ne = dxe+ n




Properties






x − 1 < bxc ≤ x ≤ dxe < x + 1;


dx + ne = dxe+ n




Properties






x − 1 < bxc ≤ x ≤ dxe < x + 1;

b−xc = −dxe;

d−xe = −bxc;bx + nc = bxc+ n;

dx + ne = dxe+ n




Properties






x − 1 < bxc ≤ x ≤ dxe < x + 1;

b−xc = −dxe;d−xe = −bxc;

bx + nc = bxc+ n;

dx + ne = dxe+ n




Properties






x − 1 < bxc ≤ x ≤ dxe < x + 1;


dx + ne = dxe+ n




Properties






x − 1 < bxc ≤ x ≤ dxe < x + 1;


dx + ne = dxe+ n



Example

Theorm

Prove that if x is a real number, then b2xc = bxc+ bx + 12c.

Proof.

Let x = n + ε, where n is an integer, and 0 ≤ ε < 1, i.e., bxc = n.

If 0 ≤ ε < 12 . In this case, 2x = 2n + 2ε and b2xc = 2n. We

also have bx + 12c = n since 0 < ε+ 1

2 < 1. Consequently,b2xc = bxc+ bx + 1

2c.If 1

2 ≤ ε < 1. In this case, 2x = 2n + 2ε = (2n + 1) + (2ε− 1)since 0 ≤ 2ε− 1 < 1, i.e., b2xc = 2n + 1. Becausebx + 1

2c = bn + 1 + (ε− 12)c = n + 1 since 0 ≤ ε− 1

2 < 1.Therefore, b2xc = 2n + 1 = n + (n + 1) = bxc+ bx + 1

2c.



Example

Theorm


Proof.





2c.

If 12 ≤ ε < 1. In this case, 2x = 2n + 2ε = (2n + 1) + (2ε− 1)

since 0 ≤ 2ε− 1 < 1, i.e., b2xc = 2n + 1. Becausebx + 1

2c = bn + 1 + (ε− 12)c = n + 1 since 0 ≤ ε− 1


2c.



Example

Theorm


Proof.





2c.If 1

2 ≤ ε < 1. In this case, 2x = 2n + 2ε = (2n + 1) + (2ε− 1)since 0 ≤ 2ε− 1 < 1, i.e., b2xc = 2n + 1. Becausebx + 1

2c = bn + 1 + (ε− 12)c = n + 1 since 0 ≤ ε− 1


2c.



Some important functions——Indicator function

Definition

The indicator function of a subset A of set S is a function

IA : X → {0, 1},

defined as

IA(x) :=

{1, if x ∈ A;0, else.

Example

A = {(x , y) : x2 + y2 ≤ 1 ∧ x ≥ 0 ∧ y ≥ 0}, andS = {(x , y) : 1 ≥ x ≥ 0 ∧ 1 ≥ y ≥ 0}∀Pi ∈ S , we define IA(Pi ) and IS−A(Pi );

π ≈ 4∑n

i=1 IA(Pi )∑ni=1 IA(Pi )+

∑ni=1 IS−A(Pi )




Definition


IA : X → {0, 1},

defined as

IA(x) :=

{1, if x ∈ A;0, else.

Example


π ≈ 4∑n


∑ni=1 IS−A(Pi )




Definition


IA : X → {0, 1},

defined as

IA(x) :=

{1, if x ∈ A;0, else.

Example


π ≈ 4∑n


∑ni=1 IS−A(Pi )



Set covering problem

Input

Universal set U ={u1, u2, · · · , un}Subsets S1,S2, · · · , Sm ⊆ UCost c1, c2, · · · , cm

Goal

Find a set S = {Si :i ∈ I} that minimizes∑

i∈I ci ,such that

⋃i∈I Si = U

Solution

Let xi = ISi =

{1, if Si ∈ S;0, else.

, yij = Iui =

{1, if ui ∈ Sj ;0, else.

Objective: min∑m

j=1 xjs.t.

∑mj=1 xjyij ≥ 1.

xi ∈ {0, 1}, and yij ∈ {0, 1}where 1 ≤ j ≤ m and 1 ≤ i ≤ n



Set covering problem

Input

Universal set U ={u1, u2, · · · , un}Subsets S1,S2, · · · , Sm ⊆ UCost c1, c2, · · · , cm

Goal

Find a set S = {Si :i ∈ I} that minimizes∑

i∈I ci ,such that

⋃i∈I Si = U

Solution

Let xi = ISi =

{1, if Si ∈ S;0, else.

, yij = Iui =

{1, if ui ∈ Sj ;0, else.

Objective: min∑m

j=1 xjs.t.

∑mj=1 xjyij ≥ 1.

xi ∈ {0, 1}, and yij ∈ {0, 1}where 1 ≤ j ≤ m and 1 ≤ i ≤ n



Some important functions——Sigmoid functions

Definition

A sigmoid function is a mathematical function having a characteristic“S”-shaped curve or sigmoid curve. For example S(x) = 1

1+e−x .

Logistic function

Graphs General form

f (x) =L

1 + e−k(x−x0),

where x0 is the x-value of the sig-moid’s midpoint, L is the maximumvalue, and k is the steepness of thecurve.




Definition

A sigmoid function is a mathematical function having a characteristic“S”-shaped curve or sigmoid curve. For example S(x) = 1

1+e−x .

Logistic function

Graphs General form

f (x) =L

1 + e−k(x−x0),

where x0 is the x-value of the sig-moid’s midpoint, L is the maximumvalue, and k is the steepness of thecurve.




Applications

Binary classification in logistic regression model;

Activation function in artificial neurons;

Cumulative distribution function in statistics.

Other forms


Take-aways

Take-aways

Conclusions

Function


Inverse functions and compositions of functions



Documents

Discrete Mathematics and Its Applicationsdase.ecnu.edu.cn/mgao/teaching/DM_2020_Spring/slides/2_sets23.pdf · Let A and B be nonempty sets. A function f from A to B is an assignment