CPT107-Discrete Mathematics and Statistics 课程笔记

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1. The most basic datatypes

1.1 The Natural Numbers

N = { 0 , 1 , 2 , 3 , … } \Bbb{N}=\{0, 1, 2, 3,\dots\} N={0,1,2,3,…}
We say that $\mathbb{N}$ is closed under addition because if you take any two numbers in $\mathbb{N}$ and add them, you always get another number in $\mathbb{N}$.

Key property : Any natural number can be obtained from $0$ by applying the operation $S(n)=n+1$ some number times.

Examples : $S(0)=1,S(S(0))=2,S(S(S(0)))=3$

1.2 The Integers

Z = { 0 , 1 , 2 , 3 , … } \Bbb{Z}=\{0, 1, 2, 3,\dots\} Z={0,1,2,3,…}
And the positive integers:
Z + = { 0 , 1 , 2 , 3 , … } \Bbb{Z}^+=\{0, 1, 2, 3,\dots\} Z+={0,1,2,3,…}

1.3 The Rational Numbers

$\mathbb{Q}$ is the set of all numbers that can be written as $\frac{x}{y}$ where $x$ and $y$ are integers and $y$ is not $0$.

1.4 The Real Numbers

$\mathbb{R}$ is the set of all numbers ---- distances to points on a number line.
A real number that is not rational is called irrational.

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2. Proof Techniques

2.1 Finding a counter-example

e.g.
statement : Let $f(n)=n^2+n+41$. For every natural number $n$, $f(n)$ is prime.
counter-example : $f(40)$

2.2 Proof by contradiction

e.g.
statement : $\sqrt{2}$ is not a rational number
Proof by contradiction :

1. If $\sqrt{2}$ were rational then we could write it as $\sqrt{2}=\frac{x}{y}$ where $x$ and $y$ are integers and $y$ is not $0$
2. By repeatedly cancelling common factors, we can make sure that $x$ and $y$ have no common factors so they are not both even
3. Then $2=\frac{x^2}{y^2}$ so $x^2=2y^2$ so $x^2$ is even. This means $x$ is even, because the square of any odd number is odd
4. Let $x=2w$ for some integer $w$
5. Then $x^2=4w^2$ so $2w^2=y^2$, so $y^2$ is even so $y$ is even
6. This contradicts the fact that $x$ and $y$ are not both even, so our original assumption must have been wrong

2.3 Proof by induction

e.g.
statement : For every integer $n\ge 3$, $n^2\ge 3n$
Proof by Induction :

1. Base Case : Take $n=3$. Then $3^2\ge 3\times 3$
2. Inductive Step : Assume that the statement is true for $n=m$ for $m\ge 3$ so $m^2\ge 3m$. Now consider $n=m+1$. We must show that $(m+1{)}^2\ge 3(m+1)$.
3. $(m+1{)}^2=m^2+2m+1\ge 3m+2m+1$ and since $m\ge 1$, $3m+2m+1\ge 3(m+1)$

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3. Set Theory

3.1 Notation

The union of two sets:

A ∪ B = { x ∣ x ∈ A or  x ∈ B } A\cup B=\{x|x\in A~\text{or}~x\in B\} A∪B={x∣x∈A or x∈B}

The intersection of two sets

A ∩ B = { x ∣ x ∈ A and  x ∈ B } A \cap B=\{x|x\in A~\text{and}~x\in B\} A∩B={x∣x∈A and x∈B}

The relative complement

A − B = { x ∣ x ∈ A and  x ∉ B } A-B=\{x|x\in A~\text{and}~x\notin B\} A−B={x∣x∈A and x∈/​B}

The complement

∼ A = { x ∣ x ∉ A } = U − A \sim A=\{x|x\notin A\}=U-A ∼A={x∣x∈/​A}=U−A

The symmetric difference

A Δ B = { x ∣ ( x ∈ A and  x ∉ B ) or  ( x ∉ A and  x ∈ B ) } A\Delta B=\{x|(x\in A~\text{and}~x\notin B)~\text{or}~(x\notin A~\text{and}~x\in B)\} AΔB={x∣(x∈A and x∈/​B) or (x∈/​A and x∈B)}

3.2 The algebra of sets

$A\cup B=B\cup A$
$A\cap B=B\cap A$
$A\cup (B\cup C)=(A\cup B)\cup C$
$A\cap (B\cap C)=(A\cap B)\cap C$
$A\cup \varnothing =A$
$A\cap \varnothing =\varnothing$
$A\cup U=U$
$A\cap U=A$
$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
$A\cup \sim A=U$
$A\cap \sim A=\varnothing$
$\sim U=\varnothing$
$\sim \varnothing =U$
$\sim (\sim A)=A$
$\sim (A\cup B)=\sim A\cap \sim B$
$\sim (A\cap B)=\sim A\cup \sim B$

The following statements hold

• ∅ ∈ { ∅ } \empty\in\{\empty\} ∅∈{∅} but $\varnothing \notin \varnothing$
• ∅ ⊆ { 5 } \empty\subseteq\{5\} ∅⊆{5}
• { 2 } ⊈ { { 2 } } \{2\}\nsubseteq\{\{2\}\} {2}⊈{{2}} but { 2 } ∈ { { 2 } } \{2\}\in\{\{2\}\} {2}∈{{2}}
• { 3 , { 3 } } ≠ { 3 } \{3,~\{3\}\}\ne\{3\} {3, {3}}​={3}

The Power Set

The power set $Pow(A)$ of a set $A$ is the set of all subsets of A
P o w ( A ) = { C ∣ C ⊆ A } Pow(A)=\{C|C\subseteq A\} Pow(A)={C∣C⊆A}

$Pow(A\cap B)=Pow(A)\cap Pow(B)$
$Pow(A\cup B)\ne Pow(A)\cup Pow(B)$

3.3 Cardinality of sets

The cardinality of a finite set S is the number of elements in S, and is denoted by |S|

$|A\cup B|=|A|+|B|-|A\cap B|$

3.4 Ordered pairs

In discussing sets, the order in which elements are listed is unimportant. In order to handle ordered lists of objects we first introduce :
The cartesian product $A\times B$ of sets $A$ and $B$ is the set consisting of all pairs $(a,b)$ with $a\in A$ and $b\in B$
A × B = { ( a , b ) ∣ a ∈ A and  b ∈ B } A\times B=\{(a,b)|a\in A~\text{and}~b\in B\} A×B={(a,b)∣a∈A and b∈B}
The cartesian product $A_1\times A_2\times \cdots \times A_n$ is :
A 1 × A 2 × ⋯ × A n = { ( a 1 , … , a n ) ∣ a 1 ∈ A 1 , … , a n ∈ A n } A_1\times A_2\times\cdots\times A_n=\{(a_1,\dots,a_n)|a_1\in A_1,\dots,a_n\in A_n\} A1​×A2​×⋯×An​={(a1​,…,an​)∣a1​∈A1​,…,an​∈An​}

$(a,b)=(c,d)$ iff $a=c\text{and}b=d$
{ 1 , 2 } = { 2 , 1 } \{1,2\}=\{2,1\} {1,2}={2,1} but $(1,2)\ne (2,1)$

e,g,
Let A = { 1 , 2 } A=\{1,2\} A={1,2} and B = { a , b , c } B=\{a,b,c\} B={a,b,c}. Then
A × B = { ( 1 , a ) , ( 2 , a ) , ( 1 , b ) , ( 2 , b ) , ( 1 , c ) , ( 2 , c ) } A\times B=\{(1,a),~(2,a),~(1,b),~(2,b),~(1,c),~(2,c)\} A×B={(1,a), (2,a), (1,b), (2,b), (1,c), (2,c)}

Cartesian plane
The set $\mathbb{R}\times \mathbb{R}$, or ${\mathbb{R}}^2$ consists of all pairs of real numbers $(x,y)$, ${\mathbb{R}}^2$ is called the Cartesian plane

Bit strings of length n

Let B = { 0 , 1 } B=\{0,1\} B={0,1}, $B^n$ consists of all lists of zeros and ones of length $n$. These are called bit strings of length n.

Bit strings can be used to represent the subsets of a set.
Suppose we have a set S = { s 1 , … , s n } S=\{s_1,\dots,s_n\} S={s1​,…,sn​}, if we have a subset $A\subseteq S$, the characteristic vector of $A$ is the n-bit string $(b_1,\dots ,b_n)\in B^n$ where $$b_i=\{\begin{array}{ll}1& \text{if}{s}_i\in A\\ 0& \text{if}{s}_i\notin A\end{array}$$

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4. Relations

4.1 Unary Relation

Unary relation is a relation in one set (actually it is just a subset of a set)

A binary relation is that it is a relation between two sets.

And there is ternary relations (relations between three sets)

4.2 Binary Relation

Binary relation : A binary relation between two sets $A$ and $B$ is a subset $R$ of the Cartesian product $A\times B$. If $A=B$, then $R$ is called a binary relation on $A$

Properties

A binary relation $R$ on a set $A$ is

• Reflexive when $xRx$ for all $x\in A$ (有向图 $A$ 内的所有节点都有自环)
• Symmetric when $xRy$ implies $yRx$ for all $x,y\in A$ ($A$ 是无向图)
• Antisymmetric : when $xRy$ and $yRx$ imply $x=y$ for all $x,y\in A$ (有向图 $A$ 内不存在双向边)
• Transitive when $xRy$ and $yRz$ imply $xRz$ for all $x,y,z\in A$ (传递性)

4.2.1 Directed Graphs (有向图)

Directed graph is a representation of binary relations. Assume that $R$ is a binary relation between $A$ and $B$. We represent the elements of these two sets as vertices of a graph, for each $(a,b)\in R$, we draw an arrow linking the related elements. This is called the directed graph (or digraph) of $R$.

4.2.2 Matrices (邻接矩阵)

A way of representing a binary relation between finite sets uses an matrix. $$M(i,j)=\{\begin{array}{ll}T& \text{if }(a_i,b_j)\in R\\ F& \text{if }(a_i,b_j)\notin R\end{array}$$

4.2.3 Transitive Closure (传递闭包)

Given a binary relation $R$ on a set $A$, the transitive closure $R^{\ast }$ of $R$ is the (uniquely determined) relation on $A$ with the following properties:

• $R^{\ast }$ is transitive
• $R\subseteq R^{\ast }$
• If $S$ is a transitive relation on $A$ and $R\subseteq S$, then $R^{\ast }\subseteq S$

传递闭包的意义

邻接矩阵是显示两点的直接关系，如 $a$ 直接能到 $b$，就为1。而传递闭包显示的是传递关系，如 $a$ 不能直接到 $c$，却可以通过 $a$ 到 $b$ 到 $d$ 再到 $c$，因此 $a$ 到 $c$ 为1。

代码实现计算传递闭包

设 $E$ 为原邻接矩阵，只需要对 Floyed 算法进行一点点改动：

for (int k = 1; k <= n; ++k)for (int i = 1; i <= n; ++i)for (int j = 1; j <= n; ++j)if (E[i][k] && E[k][j])E[i][j] = 1;

4.2.4 Equivalence Relations

A binary relation $R$ on a set $A$ is called an equivalence relation if it is reflexive, transitive, and symmetric.
The equivalence class $E_x$ of any $x\in A$ is defined by E x = { y ∣ y R x } E_x=\{y~|~yRx\} Ex​={y ∣ yRx}

仅满足自反、对称的二元关系，称为相容关系

4.2.5 Partition of a set

A partition of a set $A$ is a collection of non-empty subsets $A_1,\cdots ,A_n$ of A satisfying

• $A=A_1\cup A_2\cup \cdots \cup A_n$
• $A_i\cap A_j=\varnothing$ for $i\ne j$
  
  $A_i$ are called the blocks of the partition

Theorems

• Let $R$ be an equivalence relation on a non-empty set $A$. Then the equivalence classes { E x ∣ x ∈ A } \{E_x~|~x\in A\} {Ex​ ∣ x∈A} form a partition of A.
• Suppose that $A_1,\cdots ,A_n$ is a partition of $A$. Define a relation $R$ on $A$ by setting: $xRy$ if and only if there exists $i$ such that $1\le i\le n$ and $x,y\in A_i$. Then $R$ is an equivalence relation.

4.2.6 Partial orders

A binary relation $R$ on a set $A$ which is reflexive, transitive and antisymmetric is called a partial order

Partial orders are important in situations where we wish to characterise precedence.

Predecessor: If $R$ is a partial order on a set $A$ and $xRy$, $x\ne y$, we call $x$ is a predecessor of $y$.
If $x$ is a predecessor of $y$ and there is no z ∉ { x , y } z\notin\{x,y\} z∈/​{x,y} for which $xRz$ and $zRy$, we call $x$ an immediate predecessor of $y$

4.2.7 Hasse Diagram

The Hasse Diagram of a partial order is a digraph. The vertices of the digraph are the elements of the partial order, and the edges of the digraph are given by the “immediate predecessor” relation

4.2.8 Total orders

A binary relation $R$ on a set $A$ is a total order if it is a partial order such that any $x,y\in A,xRy$ or $yRx$

全序关系的哈斯图是一条链。

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5. Overview of functions and PHP

For a subset $y$ of $B$, the inverse of y : = f − 1 ( y ) = { x in  A ∣ f ( x ) in  y } y:=f^{-1}(y)=\{x~\text{in}~A|f(x)~\text{in}~y\} y:=f−1(y)={x in A∣f(x) in y}

There is an inverse function for $f$ if and only if $f$ is a bijection

Pigeonhole Principle

A function from a larger set to a smaller set cannot be injective.

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6. Propositional Logic

Propositional logic is concerned with

• The truth and falsity of simple (declarative) statements.
• The question: when does a statement follow from a set of statements?
• Algorithms which decide whether a statement follows from a set of statements.

6.1 Formulas

A formula is defined “syntactically”, you build a formula following rules. This has nothing to do with what the formula means (which is called “semantics”). “Parsing” the formula means determining how the formula is built, following the rules.

atomic formulas
The simplest formulas are “propositional variables” which we write using letters such as $p$, $q$ or $r$ or using indexed letters such as $p_0,p_1,p_2,\dots$
Propositional variables are called atomic formulas.

The alphabet of propositional logic consists of

• An infinite set $p,q,r,p_0,p_1,\dots$ of atomic formulas.
• The logical connectives $\neg (not),\wedge (and),\vee (or)$
• Brackets

The set $P$ of all formulas of propositional logic is defined inductively:

• All atomic formulas are formulas
• If $P$ is a formula, then $\neg P$ is a formula
• If $P$ and $Q$ are formulas, then $(P\wedge Q)$ is a formula
• If $P$ and $Q$ are formulas, then $(P\vee Q)$ is a formula
• Nothing else is a formula

Parse Trees
A parse tree is a tree representation of how the formula is constructed

• The parse tree of an atomic formula has one node, labelled with the formula.
• The parse tree of a formula $\neg P$ haa a root labelled $\neg$. The child of the root is the parse tree for the formula $P$.
• The parse tree of a formula $P\wedge Q$ has a root labelled $\wedge$. The left child of the root is the parse tree fo the formula $P$. The right child of the root is the parse tree for the formula $Q$.

Formulas are just strings over a certain alphabet without truth values or meaning.

6.2 Truth Values

An interpretation $I$ is a function which assigns to any atomic formula $p_i$ a truth value. I ( p i ) ∈ { 0 , 1 } I(p_i)\in\{0,1\} I(pi​)∈{0,1}

• If $I(p_i)=1$, then $p_i$ is called true under the interpretation $I$.
• If $I(p_i)=0$, then $p_i$ is called false under the interpretation $I$.

Given an assignment I we can compute the truth value of compound formulas step by step using so-called truth tables

Truth Tables

e.g. negation:

$I(\neg P)=\{\begin{array}{ll}0& \text{if}I(P)=1\\ 1& \text{if}I(P)=0\end{array}$

The corresponding truth table is:

6.3 Tautology

A tautology is a formula which is true under all interpretations.

$\top :=(p_1\vee \neg p_1)$

6.4 Contradiction

A contradiction is a formula which is false under all interpretations.

$\perp :=(p_1\wedge \neg p_1)$

6.5 Semantic consequence

Suppose $\Gamma$ is a finite set of formulas and $P$ is a formula. Then $P$ follows from $\Gamma$ (or is a semantic consequence of $\Gamma$) if the following implication holds for every interpretation $I$: $$\text{if}I(Q)=1\text{for all}Q\in \Gamma ,\text{then}I(P)=1$$This is denoted by $\Gamma \models P$
P 不一定在集合 $\Gamma$ 内

The following conditions are equivalent:

• { P } ⊨ Q \{P\}\vDash Q {P}⊨Q
• $(P\to Q)$ is a tautology
• $(P\wedge \neg Q)$ is a contradiction

6.5 Logical equivalence

Two formulas $P$ and $Q$ are called equivalent if they have the same truth value under every possible interpretation. In other words, $P$ and $Q$ are equivalent if $I(P)=I(Q)$ for every interpretation $I$. This is denoted by $P\equiv Q$

Laws for equivalences

• Associative laws: $(P\vee (Q\vee R))\equiv ((P\vee Q)\vee R)$
• Commutative laws: $(P\vee Q)\equiv (Q\vee P)$
• Identity laws: $(P\vee \top )\equiv P$
• Distributive laws: $(P\wedge (Q\vee R))\equiv ((P\wedge Q)\vee (P\wedge R))$
• Complement laws: $P\vee \neg P\equiv \top$
• De Morgan’s laws: $\neg (P\vee Q)\equiv (\neg P\wedge \neg Q)$

The relation $\equiv$ is an equivalence relation on $P$

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7. Predicate Logic

Limitations of Propositional Logic

Propositional logic is good for dealing with statements built from basic propositions using and, or, not, if … But more complex statements cannot be expressed naturally in propositional logic.

So we need to extend propositional logic. Numerous extensions have been developed, among them:

• Modal Logic: Adds elements to reason about necessity
• Temporal Logic: Adds elements to reason about time.
• Predicate Logic: Adds elements to reason about properties of objects

Quantifiers

• $\forall$ : for all, Universal quantification
• $\exists$ : there exists an, Existential quantification

e,g,
Every student is younger than some instructor.

this statement can be written symbolically using predicates, variables, and quantifiers:

$\forall x(Student(x)\to \exists y(Younger(x,y)\wedge Instructor(y)))$

This is a typical formula in predicate logic

Predicate logic has two additional features:

1. Equality: allows to express that individuals are equal or not
2. function symbols: allows to express functional dependencies between individuals

Function symbols can often be “simulated” by predicates, so they are not really necessary. But function symbols lead to more natural descriptions

7.1 The Language of Predicate Logic

Predicate logic allows us to express statements about

• objects
• their properties and relations to each other

Some terminologies :

• Domain : a non-empty set of objects
• Constants : concrete objects in the domain
• Functions : takes objects in the domain as arguments and returns an object of the domain
• Predicates : takes objects in the domain as arguments and returns true or false. They describe properties of objects or relationships between objects
• Variables : placeholders for concrete objects in the domain
• Quantifiers : for how many objects in the domain is the statement true

7.2 Syntax of predicate logic

Signatures

A signature defines the “vocabulary” relative to which formulae can be expressed

A signature is a set consisting of :

• predicate symbols, each with an associated arity
• function symbols, each with an associated arity
• constant symbols

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8. Combinatorics

Special Case for sums and products

$$\sum _{i\in \varnothing }f(i)=0$$
$$\prod _{i\in \varnothing }f(i)=1$$

permutations

$A_n^m=\frac{n!}{(n-k)!}$

subsets

$C_n^m=\frac{n!}{(n-k)!k!}$

The quantity which gives the number of size-k subsets of a set of size n, is called a binomial coefficient
It is writen an ${(}_k^n)$ and pronounced “n choose k”

8.1 Discrete Probability

The sample space $\Omega$ is the set of all possible outcomes. Every element $x$ of $\Omega$ has a probability $Pr(x)\in [0,1]$ $$\sum _{x\in \Omega }Pr(x)=1$$

Events

An event is a subset $X\subseteq \Omega$. The probability of the event is given by $Pr(X)={\sum }_{x\in X}Pr(x)$

$Pr(\varnothing )=0$
$Pr(\Omega )=1$

$Pr(X\cup Y)=Pr(X)+Pr(Y)-Pr(X\cap Y)$

8.2 Conditional Probability

$Pr(X|Y)=\frac{Pr(X\cap Y)}{Pr(Y)}$

8.3 Independence

Suppose that $Pr(X)\ne 0$ and $Pr(Y)\ne 0$. We say that events $X$ and $Y$, are independent if $Pr(X\cap Y)=Pr(X)Pr(Y)$

that is, $Pr(X|Y)=Pr(X)$

8.4 A random variable

A random variable is a function with domain $\Omega$

8.5 Expectation

The expected value of a random variable f is defined as follows $$E[f]=\sum _{x\in \Omega }f(x)Pr(x)$$

This is sometimes called the expectation of $f$ or the mean of $f$. It is the value that you expect to get if you pick $x$ randomly from $\Omega$ and then compute $f(x)$

If you repeatedly pick $x$ randomly from $\Omega$ and compute $f(x)$ and you compute the average of the $f(x)$ values, this tends towards $E[f]$.

Linearity of expectation

If f and g are two random variables, then $E[f+g]=E[f]+E[g]$

If $g(x)=cf(x)$, then $E[g]=cE[f]$

But $E[f\times g]=E[f]\times E[g]$ is not always true

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