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.
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}
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)...\}
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)
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
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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