• 👨‍🏫 No, that's more of an algorithm...

Other unnamed rules

• proof by contradiction $$\begin{gathered} \neg p\to F \\ --- \\ p \end{gathered}$$
• informal "Rule of Equivalence" $$\begin{gathered} p\equiv q \\ p \\ --- \\ q \end{gathered}$$

Proof

• theorem: assuming following conditions $$\begin{gathered} p\to r \\ \neg p\to q \\ q\to s \end{gathered}$$
  • wants to prove: $\neg r\to s$
• process
  1. $p\to r$ // premise
  2. $\neg r\to \neg p$ // rule of equivalence
  3. $\neg p\to q$ // premise
  4. $\neg r\to q$ // rule of syllogism 2,3
  5. $q\to s$ // premise
  6. $\neg r\to s$ // rule of syllogism 4,5
• above proof: can be checked by machine!
  • if it understands all symbols, etc.
  • 👨‍🏫 above: a "formal, mathematical" proof
• and, we can now add a new rule $$\begin{gathered} p\to r \\ \neg p\to q \\ q\to s \\ --- \\ \neg r\to s \end{gathered}$$

Predicates

• Boolean statement w/ variable
  • $p(x)$: a predicate if it becomes a prop. when $x$ is replaced by a value
    • 👨‍🏫in our UNIVERSE
• examples
  • $p(x):=x^2=2$
  • $p(5):=5^2=2$ // false
  • $p(\sqrt{2}):={\sqrt{2}}^2=2$ // true
    • however, if if out "universe" is that of integer
    • no solution for the above exists
• we can have multiple variables, too
  • $p(x,y):=x>\sqrt{y}$

Quantifiers

• universal quantifier $\forall$
  • "forall (in universe) s.t. $p(x)$"
  • $\forall xp(x)$
• existential quantifier $\exists$
  • "exists (at least one in the universe...) s.t. $p(x)$"
  • $\exists xp(x)$
• example: $\forall x{x}^2\ge 0$
  • if universe is within the $\mathbb{R}$: true
  • alternatively: $\forall x\in \mathbb{R},x^2\ge 0$

Rules on $\forall -\exists$

• $\forall -\exists$ relation
  • $\neg \forall xp(n)\equiv \exists x\neg p(n)$
  • $\exists x\neg p(n)\equiv \neg \forall xp(n)$
• ⭐distributive law
  • $\forall xp(x)\wedge q(x)\equiv \forall xp(x)\wedge \forall xq(x)$
  • $\exists xp(x)\vee q(x)\equiv \exists xp(x)\vee \exists xq(x)$
  • 👨‍🏫 but not the following!!
    • $\exists xp(x)\wedge q(x)\equiv \exists xp(x)\wedge \exists xq(x)$
    • there can be a $x$ that satisfies "either" one, but not both
    • e.g. $\exists xp(x)\wedge \neg p(x)\not\equiv \exists xp(x)\wedge \exists x\neg p(x)$
      • one on the left: w/ stricter requirement
  • also: following is wrong, for the same reason
    • $\forall xp(x)\vee q(x)\equiv \forall xp(x)\vee \forall xq(x)$
    • there can be a $x$ that only satisfies one predicate
    • e.g. $\forall xp(x)\vee \ne p(x)\not\equiv \forall xp(x)\vee \forall \neg pq(x)$
    • one on the left: w/ looser requirement here

Natural numbers

• highschool definition: $\mathbb{N}=\{0,1,2,3,\dots \}$
  • 👨‍🏫 not good! vague, what does the $\dots$ mean?
• solved by Peano, in 19th century
• Peano axioms
  1. 0 is a natural number
     • rules below: exists as equality wasn't defined by then...
     1. for every natural number $x$, $x=x$
     2. $\forall x,y\in \mathbb{N}x=y\to y=x$
     3. $\forall x,y,z\in \mathbb{N}x=y\wedge y=z\to x=z$
     4. ...
  2. every natural number $n$ has a successor $s(n)$
  3. $\forall n,m\in \mathbb{N},s(n)=s(m)\to n=m$
     • i.e. $s(n)$ is a one-to-one
  4. $\forall n\in \mathbb{N},s(n)\ne 0$
  5. if $K$ is a set s.t. $0\in K$
     • $\forall n\in \mathbb{N},n\in K\to s(n)\in K$
     • ⭐then $K\supseteq \mathbb{N}$
     • 👨‍🏫 basis of all mathematical induction
• example: I want to prove $\forall n\in \mathbb{N}p(n)$
  • $\forall n\in \mathbb{N}n\ge 0$
  • let $K$ be the set of all natural no. $n$ s.t. $p(n)$
  • showing two things are sufficient:
    • $p(0)$ holds
    • $\forall n\in \mathbb{N}p(n)\to p(n+1)$