#import "./dvd.typ": *

#show: dvdtyp.with(
  title: "Math 8 Course Notes",
  author: "Youwen Wu",
  date: "Winter 2025",
  subtitle: [Taught by Matt Porter],
  abstract: [
    In the broad light of day mathematicians check their equations and their
    proofs, leaving no stone unturned in their search for rigour. But, at night,
    under the full moon, they dream, they float among the stars and wonder at the
    miracle of the heavens. They are inspired. Without dreams there is no art, no
    mathematics, no life.
    #align(end, [-- Michael Atiyah])
  ],
)

#outline()

= Course Logistics

The textbook for the course is _Smith, Eggen, Andre. A Transition to Advanced
Mathematics. 8th ed_. #smallcaps[isbn:] `978-1-285-46326-1`. Chapters 1-5 will
be covered.

Lecture meets every M-W-F from 12:00 -- 12:50 in Phelps 1444. Recitation meets
M-W from 7:00 -- 7:50 in HSSB 1236.

== Homework

Homework is from textbook and is worth 30% of the grade, due on Gradescope.
Homework is due every W at 11:59 PM. LaTeX is recommended for typesetting but
of course we will be using Typst, the superior typesetting software for
mathematics.

Section and problem numbers should be clearly labeled and problems should be
done on a single column.

The lowest homework score will be dropped.

== Exams

Each exam is 20% of the grade. The final exam wil replace the lowest of the
first two exam scores if it is higher.

= Meeting #datetime(year: 2025, month: 1, day: 6).display()


== Trivial preliminaries

Definitions barely worth considering. Included purely for posterity.

#definition("Proposition")[
  A proposition is a sentence which is either true or false.
]

#example("Primes")[
  The numbers 5 and 7 are prime.
]

#example("Not a proposition")[
  $x^2 + 6x + 8 = 0$
]

Propositions may be stated in the formalism of mathematics using connectives,
as *propositional forms*.

#definition("Propositional forms")[
  Let $P$ and $Q$ be propositions. Then:

  + The conjunction of $P$ and $Q$ is written $P and Q$ ($P$ and $Q$).
  + The disjunction of $P$ and $Q$ is written $P or Q$ ($P$ or $Q$) (here "or" is the inclusive or).
  + The negation of $P$ is written $not P$.
]

#definition("Tautology")[
  A propositional form for which all of its values are true. In other words, a statement which is always true.
]

#definition("Contradiction")[
  A propositional form for which all of its values are false. In other words, a statement which is always false.
]

#problem[Prove that $(P or Q) or (not P and not Q)$ is a tautology][
  Trivial, omitted.
]

#example[Several denials of the statement "integer $n$ is even"][
  - It is not the case that integer $n$ is even.
  - Integer $n$ is not even.
  - $n != 2m, forall m in ZZ$
  - $n = 2m + 1, exists m in ZZ$
]

DeMorgan's Laws tell us how to distribute logical connectives across
parentheses.

#fact[DeMorgan's Laws][
  + $not (P or Q) = not P and not Q$
  + $not (P and Q) = not P or not Q$
]

#proof[
  Trivially, by completing a truth table.
]

Also, propositional forms obey commutative, associative, distributive laws,
which can be trivially obtained from symbolic manipulations and will not be
restated. Together with the double negation law and the _law of the excluded
middle_, these comprise the axioms of a system of propositional logic.

#fact[
  We abbreviate propositional forms by eliding parentheses, according to the rules:

  + $not$ is applied to the smallest proposition following it.
  + $and$ connects the smallest propositions surrounding it.
  + $or$ connects the smallest propositions surrounding it.
]

= Notes on Logic and Proofs, 1.2

_Prototypical example for this section:_ If $sin pi = 1$, then $6$ is prime.

#definition[
  For a *antedecent* $P$ and *consequent* $Q$, the *conditional sentence* $P =>
Q$ is the proposition "If $P$, then $Q$."
]

#remark[
  The statement $P => Q$ states $P$ _implies_ $Q$ and is only false if $P$ is
  true and $Q$ is false, since this is the only case where $P$ did not imply $Q$.
]

A conditional may be true even when the antedecent and consequent are unrelated.