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