#import "./dvd.typ": *

#show: dvdtyp.with(
  title: "Math 8",
  subtitle: [UC Santa Barbara],
  author: "Youwen Wu",
)

#outline()

= Chapter 1: Logic and Proofs

== 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.

#theorem[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.
]