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}

More Practice with Induction

Fibonacci Sequence

Fibonacci Sequence


Binary Relations

Let S and T be 2 sets.

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)...\}

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)


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


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

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