『Discrete Mathematics and Its Applications』离散数学及其应用学习笔记

教材是Discrete Mathematics and Its Applications 7th，做了一些简单的翻译和简单的笔记。

1. The Foundations: Logic and Proofs

ENG	CHN	Denote
negation	否定	$\neg p$
conjunction (and)	合取	$p\wedge q$
disjunction (or)	析取	$p\vee q$
exlusive or	异或	$p\oplus q$
conditional statement/ implication	蕴含	$p\to q$
biconditional statement	等价	$p\leftrightarrow q$
tautology	永真式	$\equiv 1$
contingency	可能式	$0/1$
contradiction	矛盾式	$\equiv 0$
proposition	命题	$p\to q$
converse proposition	逆命题	$q\to p$
inverse proposition	否命题	$\neg p\to \neg q$
contrapositive proposition	逆否命题	$\neg q\to \neg p$

 Precedence of Logical Operators:
 $\neg >\wedge >\vee >\to >\leftrightarrow$
 Implication Law:
 $p\to q\equiv \neg p\vee q⟺$ if $p$ then $q⟺q$ if/when $p⟺p$ only if $q⟺q$ unless $\neg p$
 Equivalence Law:
 $p\leftrightarrow q\equiv (p\to q)\wedge (q\to p)⟺p$ if and only if $q⟺p$ iff $q$

Logical Equivalences:

Equivalence	ENG	CHN
$p\wedge T\equiv p$ $p\vee F\equiv p$	Identity Laws	同一律
$p\vee T\equiv T$ $p\wedge F\equiv F$	Domination Laws	零律
$p\vee p\equiv p$ $p\wedge p\equiv p$	Idempotent Laws	幂等律
$\neg (\neg p)\equiv p$	Double Negation Law	双重否定律
$p\vee q\equiv q\vee p$ $p\wedge q\equiv q\wedge p$	Comutative Laws	交换律
$(p\vee q)\vee r\equiv p\vee (q\vee r)$ $(p\wedge q)\wedge r\equiv p\wedge (q\wedge r)$	Associative Laws	结合律
$p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)$ $p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)$	Distributive Laws	分配律
$\neg (p\wedge q)\equiv \neg p\vee \neg q$ $\neg (p\vee q)\equiv \neg p\wedge \neg q$	De Morgan’s Laws	德摩根定律
$p\vee (p\wedge q)\equiv p$ $p\wedge (p\vee q)\equiv p$	Absorption Laws	吸收律
$p\vee \neg p\equiv T$ $p\wedge \neg p\equiv F$	Negation Laws	排中律 矛盾律

 Show $p\equiv q$ :
 a. Using truth tables
 b. Using already-proved equivalences (Recommended)

Predicates and Quantifiers:

 Universal Quantifier $\forall$ :
 $\forall xP(x)⟺P(x)$ for all values of $x$ in the (restricted) domain
 Existential Quantifier $\exists$ :
 $\exists xP(x)⟺$ There exists (at least) an element $x$ in the domain such that $P(x)$
 De Morgan’s Laws for quantifiers:
 $\neg \forall xP(x)\equiv \exists x\neg P(x)$
 $\neg \exists xQ(x)\equiv \forall x\neg Q(x)$
 Notice: $\forall$ and $\exists$ have higher precedence than any logical operators
 Remark:
 All $P(x)$ are $Q(x)⟺\forall x(P(x)\to Q(x))$
 No $P(x)$ are $Q(x)⟺\forall x(P(x)\to \neg Q(x))$
 Some $P(x)$ are $Q(x)⟺\exists x(P(x)\wedge Q(x))$
 Some $P(x)$ are not $Q(x)⟺\exists x(P(x)\wedge \neg Q(x))$
 eg. Recall that the definition of $$\underset{x\to a}{\lim }f(x)=L$$  is $$\forall ϵ>0\exists \delta >0\forall x(0<|x-a|<\delta \to |f(x)-L|<ϵ)$$  And what is the inverse proposition of this statement?

Propositional Normal Forms:

ENG	CHN	Denote
Maxterm	极大项	$M=a_1\vee a_2\vee \cdots \wedge a_i\vee \cdots$ (there exists unique value of each $a_i$ so that $M=0$)
Minterm	极小项	$m=b_1\wedge b_2\wedge \cdots \wedge b_i\wedge \cdots$ (there exists unique value of each $b_i$ so that $m=1$)
Conjunctive Normal Form (CNF)	合取范式	$C=A_1\wedge A_2\wedge \cdots \wedge A_i\wedge \cdots$, $A_i$ are propositions
Disjunctive Normal Form (DNF)	析取范式	$D=B_1\vee B_2\vee \cdots \vee B_i\vee \cdots$, $B_i$ are propositions
Full Conjunctive Normal Form	主合取范式	$X=M_1\wedge M_2\wedge \cdots \wedge M_i\wedge \cdots$, $M_i$ are maxterms
Full Disjunctive Normal Form	主析取范式	$Y=m_1\vee m_2\vee \cdots \vee m_i\vee \cdots$, $m_i$ are minterms
Prenex Normal Form	前约束范式	$O=Q_1{z}_1{Q}_2{z}_2\cdots Q_n{z}_{n}E$, $Q_i=\forall$ or $\exists$, $E$ is a quantifier-free proposition

 Transforming to Prenex Normal Form:
  1.eliminate $\to$ and $\leftrightarrow$ by:
  $p\leftrightarrow q\equiv (p\to q)\wedge (q\to p)\equiv (\neg p\vee q)\wedge (\neg q\vee p)\equiv (p\wedge q)\vee (\neg p\wedge \neg q)$
  2.move all $\neg$ inward by applying De Morgan’s Laws
  3.rename variables to avoid ambiguity if necessary
  4.move all quantifiers to the front by:
  $QxA(x)\wedge P\equiv Qx(A(x)\wedge P)$,
  $QxA(x)\vee P\equiv Qx(A(x)\vee P)$,
  $Q^{\prime }xA(x)\wedge Q^{\prime \prime }yB(y)\equiv Q^{\prime }x{Q}^{\prime \prime }y(A(x)\wedge B(y))$,
  $Q^{\prime }xA(x)\vee Q^{\prime \prime }yB(y)\equiv Q^{\prime }x{Q}^{\prime \prime }y(A(x)\vee B(y))$,
  $QxQyC(x,y)\equiv QyQxC(x,y)$,
  where we have:
  $Q,Q^{\prime },Q^{\prime \prime }=\forall$ or $\exists$,
  $P$ is a quantifier-free proposition,
  $A(x),B(y),C(x,y)$ are propositions.
 *5.transform to Prenex CNF/DNF

Rules of Inference:

Rule	Tautology	ENG	CHN
$\begin{array}{rl}& p\\ & p\to q\\ \therefore & q\end{array}$	$(p\wedge (p\to q))\to q$	Modus ponens	假言推理式
$\begin{array}{rl}& \neg q\\ & p\to q\\ \therefore & \neg p\end{array}$	$(\neg q\wedge (p\to q))\to \neg p$	Modus tollens	否定拒取式
$\begin{array}{rl}& p\to q\\ & q\to r\\ \therefore & p\to r\end{array}$	$((p\to q)\wedge (q\to r))\to (p\to r)$	Hypothetical syllogism	假言三段论
$\begin{array}{rl}& p\vee q\\ & \neg p\\ \therefore & q\end{array}$	$((p\vee q)\wedge \neg p)\to q$	Disjunctive syllogism	析取三段论
$\begin{array}{rl}& p\\ \therefore & p\vee q\end{array}$	$p\to (p\vee q)$	Addition	附加律
$\begin{array}{rl}& p\wedge q\\ \therefore & q\end{array}$	$(p\wedge q)\to q$	Simplification	消去律
$\begin{array}{rl}& p\\ & q\\ \therefore & p\wedge q\end{array}$	$((p)\wedge (q))\to (p\wedge q)$	Conjunction	合取式
$\begin{array}{rl}& p\vee q\\ & \neg p\vee r\\ \therefore & q\vee r\end{array}$	$((p\vee q)\wedge (\neg p\vee r))\to (q\vee r)$	Resolution	消解规则
$\begin{array}{rl}& p\to r\\ & q\to s\\ & p\vee q\\ \therefore & r\vee s\end{array}$	$((p\to r)\wedge (q\to r)\wedge (p\vee q))\to (r\vee s)$	Constructive dilemma	构造性两难

 Remark: $(P_1\wedge P_2\wedge \cdots \wedge P_n\wedge P)\to Q\equiv (P_1\wedge P_2\wedge \cdots \wedge P_n)\to (P\to Q)$

Rule	Name
$\begin{array}{rl}& \forall xP(x)\\ \therefore & P(c)\end{array}$ , for an arbitary $c$	Universal instantiation (UI)
$\begin{array}{rl}& P(c)\\ \therefore & \forall xP(x)\end{array}$ , for an arbitary $c$	Universal generalization (UG)
$\begin{array}{rl}& \exists xP(x)\\ \therefore & P(c)\end{array}$ , for some element $c$	Existential instantiation (EI)
$\begin{array}{rl}& P(c)\\ \therefore & \exists xP(x)\end{array}$ , for some element $c$	Existential generalization (EG)

 Remark: $\exists !x\mathrm{s}.\mathrm{t}.P(x)⟺\exists x(P(x)\wedge \forall y(x\ne y\to \neg P(y)))$

2. Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

Denote	ENG
N={0,1,2,… }\mathbb{N}=\{0, 1, 2, \dots\}N={0,1,2,…}	Natural numbers
Z={…,−1,0,1,… }\mathbb{Z}=\{\dots, -1, 0, 1, \dots\}Z={…,−1,0,1,…}	Intergers
Z+={1,2,3,… }\mathbb{Z^+}=\{1, 2, 3, \dots\}Z+={1,2,3,…}	Positive integers
$\mathbb{Q}=\left\{\frac{p}{q}|p\in {\mathbb{Z}}^{+}\cup {\mathbb{Z}}^{-},q\in \mathbb{Z}\right\}$	Rational numbers
$\mathbb{R}$	Real numbers
${\mathbb{R}}^{+}$	Positve real numbers
$\mathbb{C}$	Complex numbers

ENG	CHN	Denote	i.e.
Subset	子集	$A\subseteq B$	$\forall x(x\in A\to x\in B)$
Proper subset	真子集	$A\subset B$	$\forall x(x\in A\to x\in B)\wedge \exists x(x\in B\wedge x\notin A)$
Equal	等集	$A=B$	$A\subseteq B\wedge B\subseteq A$ $\forall x(x\in A\leftrightarrow x\in B)$
Cardinality	基数	$$\begin{gathered} |S| \\ \mathrm{Card}(S) \end{gathered}$$	$$\sum _{x\in S}1$$
Power set	幂集	$\mathcal{P}(S)$	⋃A⊆S{A}\bigcup_{A \subseteq S}\{A\}A⊆S⋃​{A}

 Remark: for every set $S$, we have $\varnothing \in S$ and $S\in S$
 Cartesian products(笛卡尔积):
 A×B={(a,b) ∣ a∈A∧b∈B}A \times B=\{(a,b) \, \vert \, a \in A \land b \in B\}A×B={(a,b)∣a∈A∧b∈B}
 A1×A2×⋯×An={(a1,a2,…,an) ∣ ∀i∈{1,2,…,n}(ai∈An)}A_1 \times A_2 \times \cdots \times A_n=\{(a_1, a_2, \dots, a_n) \, \vert \, \forall i \in \{1, 2, \dots, n\}(a_i \in A_n)\}A1​×A2​×⋯×An​={(a1​,a2​,…,an​)∣∀i∈{1,2,…,n}(ai​∈An​)}

Set Operations:

Operation	CHN	Denote	i.e.
Union	并集	$A\cup B$	{x ∣ x∈A∨x∈B}\{x \, \vert \, x \in A \lor x \in B\}{x∣x∈A∨x∈B}
Intersection	交集	$A\cap B$	{x ∣ x∈A∧x∈B}\{x \, \vert \, x \in A \land x \in B\}{x∣x∈A∧x∈B}
Complement	补集	$\overline{A}$	{x∈U ∣ x∉A}\{x \in U \, \vert \, x \notin A\}{x∈U∣x∈/​A}
Difference	差集	$$\begin{gathered} A-B \\ A\setminus B \end{gathered}$$	{x ∣ x∈A∧x∉B}\{x \, \vert \, x \in A \land x \notin B\}{x∣x∈A∧x∈/​B}

 Remark:
 $A-B=A\cap \overline{B}$
 $A\cap B=\varnothing ⟺A$ and $B$ are disjoint
 Principle of Inclusion–Exclusion(容斥原理):
 $|A\cup B|=|A|+|B|-|A\cap B|$
 Symmetric difference(对称差):
 $A\oplus B=(A-B)\cup (B-A)=(A\cup B)-(A\cap B)$

Set Identities:

Identity	ENG	CHN
$$\begin{gathered} A\cup \varnothing =A \\ A\cap U=A \end{gathered}$$	Identity Laws	同一律
$$\begin{gathered} A\cup U=U \\ A\cap \varnothing =\varnothing \end{gathered}$$	Domination Laws	支配律
$$\begin{gathered} A\cup A=A \\ A\cap A=A \end{gathered}$$	Idempotent Laws	幂等律
$\overline{(\overline{A})}=A$	Complementation Law	双重互补律
$$\begin{gathered} A\cup B=B\cup A \\ A\cap B=B\cap A \end{gathered}$$	Comutative Laws	交换律
$$\begin{gathered} (A\cup B)\cup C=A\cup (B\cup C) \\ (A\cap B)\cap C=A\cap (B\cap C) \end{gathered}$$	Associative Laws	结合律
$$\begin{gathered} A\cup (B\cap C)=(A\cup B)\cap (A\cup C) \\ A\cap (B\cup C)=(A\cap B)\cup (A\cap C) \end{gathered}$$	Distributive Laws	分配律
$$\begin{gathered} \overline{A\cap B}=\overline{A}\cup \overline{B} \\ \overline{A\cup B}=\overline{A}\cap \overline{B} \end{gathered}$$	De Morgan’s Laws	德摩根定律
$$\begin{gathered} A\cup (A\cap B)=A \\ A\cap (A\cup B)=A \end{gathered}$$	Absorption Laws	吸收律
$$\begin{gathered} A\cup \overline{A}=U \\ A\cap \overline{A}=\varnothing \end{gathered}$$	Complement Laws	互补律

Functions:

 Let $A,B\ne \varnothing$, we define a mapping $f$ from $A$ to $B$:$$f:A\mapsto B$$ i.e.$$\forall a(a\in A\to \exists !b(b\in B\wedge f(a)=b))$$

Denote	ENG	CHN
$A$	domain	定义域
$B$	codomain	上域
$f(A)$	range	值域
$b$	image	像
$a$	preimage	原像

 Remark:f(A)={f(x) ∣ ∀x∈A}⊆Bf(A)=\{f(x) \, \vert \, \forall x \in A\} \subseteq Bf(A)={f(x)∣∀x∈A}⊆B

ENG	CHN	Denote
Injective/ One-to-one	单射	$$\begin{gathered} \forall a\in A\forall b\in A(f(a)=f(b)\to a=b) \\ ⟺\forall a\in A\forall b\in A(a\ne b\to f(a)\ne f(b)) \end{gathered}$$
Surjective/ Onto	满射	$\forall b\in B\exists a\in A(f(a)=b)$
Bijective/ One-to-one correspondent	双射	both injective and surjective
(strictly) Increasing	(严格)单调递增	$$\begin{gathered} \forall x\in D\forall y\in D(x<y\to f(x)\le f(y)) \\ \forall x\in D\forall y\in D(x<y\to f(x)<f(y)) \end{gathered}$$
(strictly) Decreasing	(严格)单调递减	$$\begin{gathered} \forall x\in D\forall y\in D(x<y\to f(x)\ge f(y)) \\ \forall x\in D\forall y\in D(x<y\to f(x)>f(y)) \end{gathered}$$
Inverse function	反函数	$$\begin{gathered} \forall y\in B,\exists !x\in A\mathrm{s}.\mathrm{t}.f(x)=y \\ ⟺f^{-1}(y)=x \end{gathered}$$

 Remark: $f:A\mapsto B$ is a bijection $⟹|A|=|B|$
 Addition of Functions:
 $(f+g)(x)=f(x)+g(x)$
 Mutiplication of Functions:
 $(f\cdot g)(x)=f(x)\cdot g(x)$
 Composition of Functions:
 $(f\circ g)(x)=f(g(x))$
 Useful properties of Floor Function $f(x)=\lfloor x\rfloor$ and Ceiling Function $g(x)=\lceil x\rceil$:
 $\lfloor -x\rfloor =-\lceil x\rceil$ , $\lceil -x\rceil =-\lfloor x\rfloor$
 $x-1<\lfloor x\rfloor \le x\le \lceil x\rceil <x+1$

Useful Summation Formulea:

Sum	Closed Form
$\sum _{k=0}^{n}a{r}^k,r\ne 0,1$	$\frac{a(1-r^{n+1})}{1-r}$
$\sum _{k=1}^{n}k$	$\frac{n(n+1)}{2}$
$\sum _{k=1}^n{k}^2$	$\frac{n(n+1)(2n+1)}{6}$
$\sum _{k=1}^n{k}^3$	$\frac{n^2(n+1{)}^2}{4}$
$\sum _{k=0}^{\infty }{x}^k,|x|<1$	$\frac{1}{1-x}$
$\sum _{k=1}^{\infty }k{x}^{k-1},|x|<1$	$\frac{1}{(1-x{)}^2}$

Cardinality of infinite sets:

 Definition: infinite set $A$ is countable $⟺|A|=|{\mathbb{Z}}^{+}|={\aleph }_0$
 Remark:
  $|{\mathbb{Q}}^{+}|=|{\mathbb{Z}}^{+}\times {\mathbb{Z}}^{+}|={\aleph }_0⟸|{\mathbb{Q}}^{+}|\le |{\mathbb{Z}}^{+}\times {\mathbb{Z}}^{+}|=|{\mathbb{Z}}^{+}|\le |{\mathbb{Q}}^{+}|$
  $|\mathbb{R}|=|(0,1)|={\aleph }_1⟸f:(-\frac{\pi }{2},\frac{\pi }{2})\mapsto \mathbb{R},f(x)=\tan (\frac{x}{\pi }+\frac{1}{2})$
  $|[0,1]|=|(0,1)|={\aleph }_1⟸|(0,1)|\le |[0,1]|=|[\frac{1}{4},\frac{3}{4}]|\le |(0,1)|$

 Cantor’s Theorem(康托尔定理) and proof: $|\mathcal{P}(A)|>|A|$
  Let $f:A\mapsto \mathcal{P}(A)$ be an arbitary function.
  Consider B={x∈A:x∉f(x)}B=\{x \in A : x \notin f(x)\}B={x∈A:x∈/​f(x)}.
  Assume that there exists $y\in A$ so that $f(y)=B$.
  If $y\in B$ then $y\in f(y)$, conflicting with the definition of $B$.
  If $y\notin B$ then $y\notin f(y)$, so we have $y\in B$ conflicting with $y\notin B$.
  Thus there is no such $y$, showing $B\in \mathcal{P}(A)$ does not have preimage in $A$.
  Hence, $f$ is not a surjection.
  $\mathrm{Q}.\mathrm{E}.\mathrm{D}.$
 Schrőder-Bernstein Theorem(伯恩斯坦定理): $|A|\le |B|\wedge |B|\le |A|⟺|A|=|B|$
 The Continuum Hypothesis(连续统假设): $\nexists a\mathrm{s}.\mathrm{t}.{\aleph }_0<a<{\aleph }_1$

6. Counting

 The Generalized Pigeonhole Principle(鸽巢原理/抽屉原理):
  If $N$ objects are placed into $k$ boxes, then there is at least one box containing at least $\lceil \frac{N}{k}\rceil$ objects.

Permutations and Combinations:

 $P(n,r)=n(n-1)(n-2)\cdots (n-r+1)=\frac{n!}{(n-r)!}$
 $C(n,r)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{P(n,r)}{r!}=\frac{n(n-1)(n-2)\cdots (n-r+1)}{r!}=\frac{n!}{r!(n-r)!}$
 Remark: $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-r}\right)$
 The Binomial Theorem(二项式定理): $(x+y{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{n-k}{y}^k$
  $\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)=2^n⟸x=y=1$
  $\sum _{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})=0⟸x=-y=1$
  $\sum _{k=0}^n{2}^k\left(\genfrac{}{}{0ex}{}{n}{k}\right)=3^n⟸x+1=y=2$
 Pascal’s Identity: $\left(\genfrac{}{}{0ex}{}{n+1}{k}\right)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)+\left(\genfrac{}{}{0ex}{}{n}{k-1}\right)$
 Vandermonde’s Identity and proof: $\left(\genfrac{}{}{0ex}{}{m+n}{r}\right)=\sum _{k=0}^r\left(\genfrac{}{}{0ex}{}{m}{r-k}\right)\left(\genfrac{}{}{0ex}{}{n}{k}\right)$
  $(x+y{)}^{m+n}=\cdots +(\genfrac{}{}{0ex}{}{m+n}{r})x^{m+n-r}{y}^r+\cdots$
  $\begin{array}{rl}(x+y{)}^m(x+y{)}^n& =\cdots +\left(\sum _{k=0}^r\left(\genfrac{}{}{0ex}{}{m}{r-k}\right)x^{m-r+k}{y}^{r-k}\right)\left(\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)x^{n-k}{y}^k\right)+\cdots \\ & =\cdots +\sum _{k=0}^r\left(\left(\genfrac{}{}{0ex}{}{m}{r-k}\right)x^{m-r+k}{y}^{r-k}\cdot \left(\genfrac{}{}{0ex}{}{n}{k}\right)x^{n-k}{y}^k\right)+\cdots \\ & =\cdots +\sum _{k=0}^r\left(\genfrac{}{}{0ex}{}{m}{r-k}\right)\left(\genfrac{}{}{0ex}{}{n}{k}\right)x^{m+n-r}{y}^r+\cdots \end{array}$
  Corollary: $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{n-k}\right)\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2⟸m=r=n$
 Remark: $\left(\genfrac{}{}{0ex}{}{n+1}{r+1}\right)=\sum _{k=r}^n\left(\genfrac{}{}{0ex}{}{k}{r}\right)$
  $\begin{array}{rl}\left(\genfrac{}{}{0ex}{}{n+1}{r+1}\right)& =\left(\genfrac{}{}{0ex}{}{n}{r+1}\right)+\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n-1}{r+1}\right)+\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)+\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\cdots \\ & =\left(\genfrac{}{}{0ex}{}{r+1}{r+1}\right)+\sum _{k=r+1}^n\left(\genfrac{}{}{0ex}{}{k}{r}\right)=\left(\genfrac{}{}{0ex}{}{r}{r}\right)+\sum _{k=r+1}^n\left(\genfrac{}{}{0ex}{}{k}{r}\right)=\sum _{k=r}^n\left(\genfrac{}{}{0ex}{}{k}{r}\right)\end{array}$

Stirling Number(II):

 $n$ distinguishable objects into $k$ indistinguishable boxes: $\sum _{j=1}^{k}S(n,j)=\sum _{j=1}^k\left[\frac{1}{j!}\sum _{i=0}^{j-1}(-1{)}^i(\genfrac{}{}{0ex}{}{j}{i})(j-i{)}^n\right]$

8. Advanced Counting Techniques

Solving Linear Recurrence Relations:

 $a_n=c_1{a}_{n-1}+c_2{a}_{n-2}+\cdots +c_k{a}_{n-k}+F(n)$, $F(n)=(b_t{n}^t+b_{t-1}{n}^{t-1}+\cdots +b_{1}n+b_0)s^n$.
 Let the form of general solutions be $a_n^{(p)}+a_n^{(h)}$
 $a_n^{(h)}$ is the solution of homogeneous equation $a_n=c_1{a}_{n-1}+c_2{a}_{n-2}+\cdots +c_k{a}_{n-k}$
 Assume that $s$ is a root of multiplicity $m$ of the equation
 Then $a_n^{(p)}$is the form as $n^m(d_t{n}^t+d_{t-1}{n}^{t-1}+\cdots +d_{1}n+d_0)s^n$.
 Particularly, when $s$ is not a root of the homogeneous equation, $m=0$.
 Now we can solve the coefficients $e_i$ of $a_n=e_1{x}_1^n+e_2{x}_2^n+\cdots +e_k{x}_k^n+a_n^{(p)}$.
 If $x_i$ is of multiplicity $l$, the form is as $(e_i{n}^l+e_{i+1}{n}^{l-1}+\cdots +e_{i+l-2}n+e_{i+l-1})x_i^n$.

Generating Functions:

 $G(x)=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n+\cdots =\sum _{k=0}^{\infty }{a}_k{x}^k$
 Remark:
  $f(x)+g(x)=\sum _{k=0}^{\infty }(a_k+b_k)x^k$
  $\alpha \cdot f(x)=\sum _{k=0}^{\infty }\alpha \cdot a_k{x}^k$
  $x\cdot f^{\prime }(x)=\sum _{k=0}^{\infty }k\cdot a_k{x}^k$
  $f(\alpha x)=\sum _{k=0}^{\infty }{\alpha }^k\cdot a_k{x}^k$
  $f(x)g(x)=\sum _{k=0}^{\infty }(\sum _{j=0}^k{a}_j{b}_{k-j})x^k$
 eg.
  $b_k=\sum _{i=0}^k{a}_i$
  $⟹F(x)=\sum _{k=0}^{\infty }{b}_k{x}^k=\sum _{k=0}^{\infty }\left(\sum _{i=0}^k{a}_i\cdot 1\right)x^k=G(x)\cdot \frac{1}{1-x}$
  $a_k=k^2$
  $⟹G(x)=\sum _{k=0}^{\infty }{k}^2{x}^k=x{\left(\sum _{k=0}^{\infty }k{x}^k\right)}^{\prime }=x{\left[x{\left(\sum _{k=0}^{\infty }{x}^k\right)}^{\prime }\right]}^{\prime }=x{\left[x{\left(\frac{1}{1-x}\right)}^{\prime }\right]}^{\prime }=\frac{x(1+x)}{(1-x{)}^3}$
  $a_k=\sum _{i=0}^k{i}^2$
  $⟹G(x)=\sum _{k=0}^{\infty }(\sum _{i=0}^k{i}^2\cdot 1)x^k=\frac{x(1+x)}{(1-x{)}^3}\cdot \frac{1}{1-x}=\frac{x(1+x)}{(1-x{)}^4}$
  $f(x)=\frac{1}{1-4x^2}=\frac{1}{(1-2x)(1+2x)}=\frac{1}{2}\left(\frac{1}{1-2x}+\frac{1}{1+2x}\right)=\sum _{k=0}^{\infty }{a}_k{x}^k$
  $⟹a_k=\frac{2^k+(-2{)}^k}{2}$
 The Extended Binomial Therom:
  $(1+x{)}^u=\sum _{k=0}^{\infty }(\genfrac{}{}{0ex}{}{u}{k})x^k,|x|<1,u\in \mathbb{R}$
  Collary:
  $(1+x{)}^{-n}=\sum _{k=0}^{\infty }(\genfrac{}{}{0ex}{}{-n}{k})x^k=\sum _{k=0}^{\infty }(-1{)}^k\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)x^k$
  $(1-x{)}^{-n}=\sum _{k=0}^{\infty }(\genfrac{}{}{0ex}{}{-n}{k})(-x{)}^k=\sum _{k=0}^{\infty }\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)x^k$

Generating Functions of Common Sequences:

Sequence	Generating Function
$1$	$\frac{1}{1-x}$
${\alpha }^k$	$\frac{1}{1-\alpha x}$
$k+1$	$\frac{1}{(1-x{)}^2}$
$\left(\genfrac{}{}{0ex}{}{n}{k}\right)$	$(1+x{)}^n$
$\left(\genfrac{}{}{0ex}{}{n}{k}\right){\alpha }^k$	$(1+\alpha x{)}^n$
$\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)$	$(1-x{)}^{-n}$
$(-1{)}^k(\genfrac{}{}{0ex}{}{n+k-1}{k})$	$(1+x{)}^{-n}$
$\frac{1}{k!}$	$e^x$
$\frac{(-1{)}^{k+1}}{k!}$	$\ln (1+x)$

9. Relations

 Definition: A binary relation $R$ from set $A$ to set $B$ is a subset of $A\times B$.
 Remark: R={(a,b) ∣ a∈A∧b∈B∧aRb}⊆A×BR=\{(a,b) \, | \, a \in A \land b \in B \land aRb\} \subseteq A \times BR={(a,b)∣a∈A∧b∈B∧aRb}⊆A×B
 Definition: A relation on set $A$ is a binary relation from set $A$ to set $A$.
 eg. How many binary relations are there on a set $A$ with $n$ elements?
  $|A|=n⟹|A\times A|=n\cdot n=n^2⟹|R|=2^{n^2}$

Connection Matrices

 Let $R$ be a relation from A={a1,a2,…,an}A=\{a_1, a_2, \dots , a_n\}A={a1​,a2​,…,an​} to B={b1,b2,…,bm}B=\{b_1, b_2, \dots , b_m\}B={b1​,b2​,…,bm​}.
 The $n\times m$ connection matrix $M_R=[m_{ij}]$ for $R$ is defined by:
 $m_{ij}=\{\begin{array}{l}1,(a_i,b_j)\in R\\ 0,(a_i,b_j)\notin R\end{array}$

Special Properties of Binary Relations

Type	CHN	Denote
Reflexive	自反性	$\forall x(x\in A\to (x,x)\in R)$
Irreflexive	非自反性	$\forall x(x\in A\to (x,x)\notin R)$
Symmetric	对称性	$\forall x\forall y((x,y)\in R\to (y,x)\in R)$
Antisymmetric	反对称性	$\forall x\forall y((x,y)\in R\wedge (y,x)\in R\to x=y)$ $\forall x\forall y((x,y)\in R\wedge x\ne y\to (y,x)\notin R)$
Asymmetric	非对称性	$\forall x\forall y(x\ne y\to (x,y)\in R\wedge (y,x)\notin R\vee (x,y)\notin R\wedge (y,x)\in R)$
Transitive	传递性	$\forall x\forall y\forall z((x,y)\in R\wedge (y,z)\in R\to (x,z)\in R)$

 Remark: $\overline{(m_{ij}\wedge m_{jk})}\vee m_{ik}=1$
 eg.
  How many relations on a set with $n$ elements that are reflexive? $2^{n^2-n}$
  How many relations on a set with $n$ elements that are symmetric? $2^n\cdot 2^{\frac{n^2-n}{2}}=2^{\frac{n^2+n}{2}}$
  How many relations on a set with $n$ elements that are antisymmetric? $2^n\cdot 3^{\frac{n^2-n}{2}}$
  How many relations on a set with $n$ elements that are reflexive and symmetric? $2^{\frac{n^2-n}{2}}$
  How many relations on a set with $n$ elements that are transitive? $\sum _{j=1}^{n}S(n,j)$

Combining Relations

 Let A={a1,a2,…,an}A=\{a_1, a_2, \dots , a_n\}A={a1​,a2​,…,an​}, B={b1,b2,…,bm}B=\{b_1, b_2, \dots , b_m\}B={b1​,b2​,…,bm​}, $M_{R_1}=[c_{ij}]$, $M_{R_2}=[d_{ij}]$.

Operation	Denote
$R_1\cup R_2$	$M_{R_1\cup R_2}=[c_{ij}\vee d_{ij}]$
$R_1\cap R_2$	$M_{R_1\cap R_2}=[c_{ij}\wedge d_{ij}]$
$\overline{R_1}$	$M_{\overline{R_1}}=[\overline{c_{ij}}]$
$R_1-R_2$ $R_1\cap \overline{R_2}$	$M_{R_1-R_2}=M_{R_1\cap \overline{R_2}}=[c_{ij}\wedge \overline{d_{ij}}]$

 Let R={(a,b) ∣ a∈A∧b∈B∧aRb}R=\{(a,b) \, | \, a \in A \land b \in B \land aRb\}R={(a,b)∣a∈A∧b∈B∧aRb}, S={(b,c) ∣ b∈B∧c∈C∧bSc}S=\{(b,c) \, | \, b \in B \land c \in C \land bSc\}S={(b,c)∣b∈B∧c∈C∧bSc}.
 The composition of $R$ and $S$: S∘R={(a,c) ∣ a∈A∧c∈C∧∃b∈B(aRb∧bSc)}S \circ R=\{(a,c) \, | \, a \in A \land c \in C \land \exists b \in B(aRb \land bSc)\}S∘R={(a,c)∣a∈A∧c∈C∧∃b∈B(aRb∧bSc)}
 Remark: $R^n=R^{n-1}\circ R$
 Theroem: The relation R on a set A is transitive $⟺R^n\subseteq R$
 Let R={(a,b) ∣ a∈A∧b∈B∧aRb}R=\{(a,b) \, | \, a \in A \land b \in B \land aRb\}R={(a,b)∣a∈A∧b∈B∧aRb}.
 The inverse relation of $R$: Rc=R−1={(b,a) ∣ a∈A∧b∈B∧aRb}R^c=R^{-1}=\{(b,a) \, | \, a \in A \land b \in B \land aRb\}Rc=R−1={(b,a)∣a∈A∧b∈B∧aRb}
 Remark: $M_{R^{-1}}=(M_R{)}^T$

Properties of Relation Operations

 $(R\cup S{)}^{-1}=R^{-1}\cup S^{-1}$
 $(R\cap S{)}^{-1}=R^{-1}\cap S^{-1}$
 $(\overline{R}{)}^{-1}=\overline{R^{-1}}$
 $(R-S{)}^{-1}=R^{-1}-S^{-1}$
 $(A\times B{)}^{-1}=B\times A$
 $\overline{R}=A\times B-R$
 $(S\circ T{)}^{-1}=T^{-1}\circ S^{-1}$
 $(R\circ T)\circ P=R\circ (T\circ P)$
 $(R\cup S)\circ T=(R\circ T)\cup (S\circ T)$

Closures of Relations

Closure	CHN	Denote
Reflexive Closure	自反闭包	$r(R)=R\cup I_A$
Symmetric Closure	对称闭包	$s(R)=R\cup R^{-1}$
Transitive Closure	传递闭包	$t(R)=R^{\ast }$

 The diagonal relation on $A$: IA={(x,x) ∣ x∈A}I_A=\{(x,x) \, | \, x \in A\}IA​={(x,x)∣x∈A}
 The connectivity relation on $A$: $R^{\ast }=\bigcup _{n=1}^{\infty }{R}^n$
 Collary: $|A|=n⟹t(R)=R\cup R^2\cup \cdots \cup R^n$

Equivalence Relations and Partitions

 Definition:
  A relation $R$ on set $A$ is an equivalence relation iff $R$ is reflexive, symmetric and transitive.
  The equivalence class of $x\in A$ via equivalence relation $R$ is $[x{]}_R$, or $[x]$ for short.
  $a$ and $b$ of set $A$ are equivalent by equivalence relation $R$ $⟺a\sim b$
 Remark:
  Congruence Modulo $m$: R={(a,b) ∣ a≡b(modm),a,b∈Z}R=\{(a,b) \, | \, a \equiv b \pmod{m}, a,b \in \mathbb{Z}\}R={(a,b)∣a≡b(modm),a,b∈Z}
  Congruence class Modulo $m$: [t]m={t+km ∣ k∈Z},t=0,1,2,…,m−1[t]_m=\{t+km \, | \, k \in \mathbb{Z}\}, t=0,1,2,\dots,m-1[t]m​={t+km∣k∈Z},t=0,1,2,…,m−1
 Theroem:
  Let $R$ be an equivalence relation on set $A$.
  $aRb⟺[a]=[b]⟺[a]\cap [b]\ne \varnothing$
 Theroem:
  Let $R_1$ and $R_2$ be equivalence relations on $A$.
  $R_1\cap R_2$ is an equivalence relation, $R_1\cup R_2$ is reflexive and symmetric.
  Collary: $(R_1\cup R_2{)}^{\ast }$ is an equivalence relation
 Definition:
  A partition of set $A$ is a collection of disjoint nonempty subsets of $A$ that have $A$ as their union.
  pr(A)={Ai ∣ i∈I}pr(A)=\{A_i \, | \, i \in I\}pr(A)={Ai​∣i∈I}, where $I$ is an index set, $A_i\ne \varnothing$, $A_i\cap A_j=\varnothing (i\ne j),\bigcup _{i\in I}{A}_i=A$.

Partial Orderings

 Definition:
  A relation $R$ on set $S$ is a partial ordering iff $R$ is reflexive, antisymmetric and transitive.
  For notation, $(S,R)$ is a partially ordered set, or poset for short.
  The elements $a$ and $b$ of a poset $(S,\preccurlyeq )$ are comparable if either $a\preccurlyeq b$ or $b\preccurlyeq a$.
  When neither $a\preccurlyeq b$ or $b\preccurlyeq a$, then $a$ and $b$ are called incomparable.
  If $(S,\preccurlyeq )$ is a poset and $\forall (a,b)\in S$ are comparable, $S$ is totally ordered or linearly ordered.
  In this case, $\preccurlyeq$ is called a total order or linear order.

ENG	CHN	Denote
Maximal Element(s)	极大值	$\exists a\in A(\neg \exists b\in A(a\prec b))$
Minimal Element(s)	极小值	$\exists a\in A(\neg \exists b\in A(b\prec a))$
Greatest Element	最大值	$\exists a\in A(\forall b\in A(b\preccurlyeq a))$
Least Element	最小值	$\exists a\in A(\forall b\in A(a\preccurlyeq b))$
Upper Bound	上界	$\exists a\in S(\forall b\in A(b\prec a))$ $A\subseteq S$
Lower Bound	下界	$\exists a\in S(\forall b\in A(a\prec b))$ $A\subseteq S$
Least Upper Bound	上确界	$\min a$
Greatest Lower Bound	下确界	$\max a$

  A poset $(A,R)$ is well-ordered iff every nonempty subset of $A$ has a least element.
  A poset is called a lattice iff every pair of elements has a $\mathrm{l}\mathrm{u}\mathrm{b}$ and a $\mathrm{g}\mathrm{l}\mathrm{b}$.

10. Graphs

 Let $G=(V,E)$ be an undirected graph with $e$ edges, then $\sum _{v\in V}\mathrm{deg}(v)=2e$.
 Let $G=(V,E)$ be a digraph, then $\sum _{v\in V}{\mathrm{deg}}^{+}(v)=\sum _{v\in V}{\mathrm{deg}}^{-}(v)=|E|$.

Some Special Simple Graphs

Denote	i.e.
$K_n$	Complete Graph
$C_n(n\ge 3)$	Cycle
$W_n(n\ge 3)$	Wheel
$Q_n$	$n$-Cube
$K_{m,n}(m=|V_1|,n=|V_2|)$	Complete Bipartite Graph

 Regular graph:
  A simply graph is called regular if every vertex of this graph has the same degree.
  A regular graph is called $n$-regular if every vertex in this graph has degree $n$.

New Graphs from Old

 Subgraph:
  Let $G=(V,E)$, $H=(W,F)$.
  $H$ is a subgraph of $G$ if $W\subseteq V$ and $F\subseteq E$.
  Subgraph $H$ is a proper subgraph of $G$ if $H\ne G$.
  $H$ is a spanning subgraph of $G$ if $W=V$ and $F\subseteq E$.

 Union:
  Let $G_1=(V_1,E_1)$, $G_2=(V_2,E_2)$, then $G_1\cup G_2=(V_1\cup V_2,E_1\cup E_2)$.

Representing Graphs

 Adjacency matrix: $A_G=[a_{ij}{]}_n$, $a_{ij}=1$ if {vi,vj}\{v_i,v_j\}{vi​,vj​} is an edge of $G$, or $a_{ij}=0$ otherwise.
 Incidence matrix: $M_G=[m_{ij}{]}_{n\times m}$, $m_{ij}=1$ if $e_j$ is incident with $v_i$, or $m_{ij}=0$ otherwise.
 Isomorphism(同构):
  Formally, two simple graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ are isomorphic.
  If and only if there is an bijection $f$ from $V_1$ to $V_2$ such that $\forall a,b\in V_1$.
  $a$ and $b$ are adjacent in $G_1$ while $f(a)$ and $f(b)$ are adjacent in $G_2$.

Connectivity

 The number of different paths of length $r$ from $v_i$ to $v_j$ is equal to the $(i,j)$-th entry of $A^r$.
 The maximally connected subgraphs of $G$ are called the connected components.
 A vertex is a cut vertex or articulation point, if removing it and its incident edges results in more connected components.
 Similarly if removal of an edge creates more components the edge is called a cut edge or bridge.
 A directed graph is strongly connected if there is a path from $a$ to $b$ and from $b$ to $a$ for all vertices $a$ and $b$ in the graph.
 The graph is weakly connected if the underlying undirected graph is connected.
 For directed graph, the maximal strongly connected subgraphs are called the strongly connected components.

Euler and Hamilton Paths

 Definition:
  A connected multigraph has an Euler circuit if and only if each of its vertices has even degree.
  A connected multigraph has an Euler path but not an Euler circuit if and only if it has exactly two vertices of odd degree.
  A Hamilton path in a graph $G$ is a path which visits every vertex in $G$ exactly once.
  A Hamilton circuit visits every vertex exactly once, except for the first vertex, which is also the end of the cycle.
 Sufficient Condition:
  Let $G$ be a simple graph with $n$ $(n\ge 3)$ vertices.
  Dirac’s Theroem(狄拉克定理) If $\underset{v\in G}{\mathrm{deg}}(v)\ge \frac{n}{2}$, then $G$ has a Hamilton circuit.
  Ore’s Theroem(奥勒定理) If $\mathrm{deg}(u)+\mathrm{deg}(v)\ge n$ $(\forall u,v\in G,\neg e_{uv})$, then $G$ has a Hamilton circuit.

Planar Graphs

 Euler’s formula:
  Let $G$ be a connected planar simple graph with $e$ edges and $v$ vertices.
  Let $r$ be the number of regions in a planar representation of $G$.
  Then we have the formula $r=e-v+2$.
 Degree of Region:
  Suppose $R$ is a region of a connected planar simple graph.
  The number of edges on the boundary of $R$ is called the Degree of $R$, denoted by $\mathrm{D}\mathrm{e}\mathrm{g}(R)$.
 Collary:
  If $G$ is a connected planar simple graph with $e$ edges and $v$ $(v\ge 3)$ vertices, then $e\le 3v-6$.
   $2e=\sum _{R_i\in G}\mathrm{D}\mathrm{e}\mathrm{g}(R_i)\ge 3r⟹r=e-v+2\le \frac{2}{3}e⟹e\le 3v-6$
  If $G$ is a connected planar simple graph, then $G$ has a vertex $v_i$ with $\mathrm{deg}(v_i)\le 5$.
   $(2e=\sum _{v_i\in V}\mathrm{deg}(v_i)\ge 6v)\wedge (e\le 3v-6\iff 2e\le 6v-12)=0$
  If G has $e$ edges and $v$ $(v\ge 3)$ vertices and no circuits of length $3$,then $e\le 2v-4$.
   $2e=\sum _{R_i\in G}\mathrm{D}\mathrm{e}\mathrm{g}(R_i)\ge 4r⟹r=e-v+2\le \frac{1}{2}e⟹e\le 2v-4$
  Generally, if every region of G has at least $k$ edges, then $e\le \frac{k(v-2)}{k-2}$.
 Homeomorphism(同胚):
  The graph $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ are called homeomorphic.
  If and only if they can be obtained from the same graph by a sequence of elementary subdivision.
 Kuratowski’s Theorem(库拉托斯基定理):
  A graph is nonplanar if and only if it contains a subgraph homeomorphic to $K_{3,3}$ or $K_5$.

(updated 2021.5.14)