Discrete Math (-OH edition)

离散数学（羟基版）

Zsir：

group theory / number theory

no constraints

Yuan Zhang、Penghui Yao：

graph theory (8 weeks & 1 quiz each)

probably hw, check-in, exams (no guarantee from Zsir)

class 01, intro (Zsir)

Mostly abstract algebra

Definitions

Binary operation: $f: G^2 \rightarrow G$

Algebraic system: $(G,\cdot)$

Closedness: $\forall x,y \in G, x\cdot y\in G$

Group: Algebraic system $(G, \cdot)$ where $G$ is a set and $\cdot$ is a binary function $f: G^2 \rightarrow G$, and:

• $\exists \text{ Identity }1 \in G \text{ s.t. } \forall a \in G, a\cdot 1 = 1\cdot a = a$
• $\forall a, b, c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
• $\forall a \in G, \exists a^{-1} \in G \text{ s.t. } a^{-1} \cdot a = a \cdot a^{-1} = 1$

Example 1: Klein 4-group

Klein bottle...

The group: $(\{1, a, b, c\}, \cdot)$

$\cdot$	$1$	$a$	$b$	$c$
$1$	$1$	$a$	$b$	$c$
$a$	$a$	$1$	$c$	$b$
$b$	$b$	$c$	$1$	$a$
$c$	$c$	$b$	$a$	$1$

Example 2:

Geometrical operation on a regular triangle: ${}_{B}\mathop{\triangle}\limits^{A}{}_{C}$

1. identity operation: ${}_{B}\mathop{\triangle}\limits^{A}{}_{C} \rightarrow {}_{B}\mathop{\triangle}\limits^{A}{}_{C}$
2. clockwise rotation by $120^{\circ}$: ${}_{B}\mathop{\triangle}\limits^{A}{}_{C} \rightarrow {}_{C}\mathop{\triangle}\limits^{B}{}_{A}$
3. counter-clockwise rotation by $120^{\circ}$: ${}_{B}\mathop{\triangle}\limits^{A}{}_{C} \rightarrow {}_{A}\mathop{\triangle}\limits^{C}{}_{B}$
4. reflection: ${}_{B}\mathop{\triangle}\limits^{A}{}_{C} \rightarrow {}_{C}\mathop{\triangle}\limits^{A}{}_{B}$
5. reflection followed by clockwise rotation: ${}_{B}\mathop{\triangle}\limits^{A}{}_{C} \rightarrow {}_{B}\mathop{\triangle}\limits^{C}{}_{A}$
6. reflection followed by counter-clockwise rotation: ${}_{B}\mathop{\triangle}\limits^{A}{}_{C} \rightarrow {}_{A}\mathop{\triangle}\limits^{B}{}_{C}$

Composition is assosiative...

Example 3:

Let $M$ be the set of $n\times n$ real non-singular matrices, then $(M, \cdot)$ is a group.

Matrix multiplication is assosiative...

Uniqueness of identity and inverse

Uniqueness: $1 \cdot 1' = ?$

Cancellation law: $a\cdot b = a\cdot c \Rightarrow b=c$, and $b\cdot a = c\cdot a \Rightarrow b=c$

We wonder if cancellation law holds, whether can we say we'll get a group.

Solution of group equation obviously exists in a group:

$x \cdot a = b \Leftrightarrow x = b \cdot a^{-1}$

$a \cdot x = b \Leftrightarrow x = a^{-1} \cdot b$

Introducing the magma: Closedness.
Introducing the semigroup: Closedness, Assosiativity.
Introducing the monoid: Closedness, Assosiativity, Identity.

If cancellation law holds in a semigroup, can we say we'll get a group?

Yes for finite semigroup.

$(\Z^+,+)$

If solution of group equation always exist in a semigroup, can we say we'll get a group?

Yes.

class 02, graph theory (Yuan Zhang)

Very important, honestly

Konig's 7-bridge problem

Only finite graphs

Graph $G$ is an ordered pair of sets $(V, E)$, where $V$ is a finite set, and $E \subseteq \binom{V}{2}$.

$V$ is called the set of vertices, $E$ is called the set of edges.

We say $v_1$ and $v_2$ is adjacent iff $\exists e=v_1v_2 \in E$.

We say $e_1$ and $e_2$ is adjacent iff $e_1\cap e_2 \neq \varnothing$.

We say $v$ is incident to edge $e$ iff $e = uv$.

directed graph: $G=(V, A)$, $A \subseteq V^2$

multigraph: $G=(V, E)$, $E$ is a multiset and $E \subseteq \binom{V}{1}\cup \binom{V}{2}$

hypergraph: $G=(V, E)$, $E\subseteq 2^V \backslash \{\varnothing\}$

Important graphs and graph classes

1. complete graph $K_n = ([n], \binom{V}{2})$

2. empty graph $E_n = ([n], \varnothing)$

3. path $P_n = ([n], \{(i, i + 1): i = 1, 2, \ldots, n - 1\})$

length of a path is $\#$ of edges on the edge

4. cycle $C_n = ([n], \{(i, i + 1): i = 1, 2, \ldots, n - 1,(n,1)\})$

5. complete bipartite graph $K_{m,n} = (A\cup B, \{xy: x\in A, y\in B\})$

triangle-free graph with most edges: $K_{\frac{n}{2},\frac{n}{2}}$

Basic parameters of a graph

• order $|G| = |V|$

• size $||G|| = |E|$

$G$ is trivial iff $||G|| < 1$

(on textbook) $G$ is trivial iff $|G| \leq 1$

• degree of $v$: $d(v) = deg(v) = |\{e | e \in E, e \text{ is incident to } v\}|$

Handshake Lemma: $2 |E| = \sum\limits_{v \in V} deg(v)$

• neighborhood of vertex $v$: $N_G(v) = \{u | uv \in E\}$

• neighborhood of set of vertex $U$: $N_G(U) = \{w | vw \in E, v \in U, w \in V\backslash U\}$

• min degree $\delta(G)$

• max degree $\Delta(G)$

• average degree $d(G) = \dfrac{\sum_{v}deg(v)}{|V|}$

• walk of length $k$: $v_0 e_0 v_1 e_1 \ldots e_{k-1} v_k$, with incidency (no no-revisit guaranteed)

• closed walk: cycle

Proposition 1.3.1: If a graph has minimal degree $\delta(G) \geq 2$, then there must be a path of length $\delta(G)$ and a cycle with $\geq \delta(G) + 1$ vertices.

proof: take the longest path and look at the first vertice of the path