Lecture 013

The Well-Ordering Property (WOP)

Corollary: if $S=\{n \in \mathbb{Z}|n \geq M\}$ for some fixed $M\in \mathbb{Z}$.

So now if we want to prove $\{n\in \mathbb{Z}^+ | (\exists x, y \in \mathbb{Z}(23x - 72y = n))\}$ by only showing that the set is non-empty by WOP.

Consequence of WOP: there exists no infinitely descending chain of natural numbers.

Proof by The Methods of Infinite Descent

essentially contradiction + WOP WTS: $(\forall n \in \mathbb{N})P(n)$ Show: $(\forall n \in \mathbb{N})(\lnot P(n) \implies (\exists k \in \mathbb{N})(k < n \land \lnot P(k)))$ Conclude: $\lnot P(n)$ never holds

Claim: $\sqrt{2}$ is irrational Proof: AFSOC $\sqrt{2} \in \mathbb{Q}$

• Then $\sqrt{2} = \frac{a_1}{b_1}$. Since $a_1^2 = 2b^b_2$, $a_1^2$ is even hence $a_1$ is even. So $a_1 = 2a_2$ for some integer a2. Then we can write $\sqrt{2} = \frac{a_1}{b_1}=\frac{2a_2}{b_1}$. Then we have $2b_1^2 4a_^2_2$ to prove that $b_1$ must be even. So $\sqrt{2} = \frac{a_1}{b_1}=\frac{2a_2}{b_1} = \frac{2a^2}{2b^2}... infinitely descent, contradiction with WOP.

More Practice with Induction

Relations

Binary Relations

Let S and T be 2 sets.

• If $R \subseteq S \times T$, then R is called binary relation between S and T.

• If $(a, b) \in R$, we say "a is in relation to b", detated aRb.

• If S is called the domain, and T is called co-domain.

• If S=T, R is a relation on S.

Example: S=students, T=teachers. define $R \subseteq S \times T$ by $(s, t) \in R \equiv$ s has taken a class with professor t.

Example: "less than" = $\{(0, 1), (0, 2), ..., (1, 2), (1, 3)...\}$

• (0, 1) is in set "less than" = 0<1.

Example: "subset or equal", let $S=T=\mathcal{P}(\mathbb{N})$. for $A\in S$ and $B \in T$, $A\subseteq B \equiv (\forall m \in \mathbb{N})(m\in A \implies m\in B)$

Definitions

Let R be a binary relation on a set S. Then R is called: (R is a set or ordered tuple)

• reflexive(R): iff $(\forall x \in S)((x,x) \in R)$ (any element is related to itself) 自己和自己比永远存在

• irreflecsive(IR): iff $(\forall x \in S)((x,x) \notin R)$ (no element is related to itself, notice it is not the opposite of relfecsive) 自己和自己比永远不存在

• symmetric(S): iff $(\forall x,y \in S)((x, y) \in R \implies (y, x) \in R) 正反永远成立

• antisymmetric(AS): iff $(\forall x,y \in S)((x, y) \in R \land (y, x) \in R \implies x = y) 正反成立则相等

• transitive(TR): iff (\forall x, y, z \in R)((x, y)\in R \land (y, z)\in R \implies (x, z) \in R) 传递

• Total(To): $(\forall x, y \in S)(x \neq y \implies ((x, y) \in R \lor (y, x) \in R))$ 两个数总有联系

Chart

Set	Relation	Ref	Irrefl	Symm	Antisymm	Trans
$\mathbb{R}$	$<$	N	Y	N	Y	Y
$\mathbb{Z}$	divisible	Y	N	N	N	Y
$\mathcal{P}(S)$	$\subseteq$	Y	N	N	Y	Y
$S$	$=$	Y	N	Y	Y	Y
$\mathbb{R}$	$\leq$	Y	N	N	Y	Y
$\mathcal{P}(S)$	$\subsetneq$	N	Y	N	Y	Y