107 lines
2.9 KiB
Text
107 lines
2.9 KiB
Text
#import "./dvd.typ": *
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#show: dvdtyp.with(
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title: "Math 8 Course Notes",
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author: "Youwen Wu",
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date: "Winter 2024",
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subtitle: [Taught by Matt Porter],
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abstract: [
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In the broad light of day mathematicians check their equations and their
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proofs, leaving no stone unturned in their search for rigour. But, at night,
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under the full moon, they dream, they float among the stars and wonder at the
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miracle of the heavens. They are inspired. Without dreams there is no art, no
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mathematics, no life.
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#align(end, [-- Michael Atiyah])
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],
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)
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#outline()
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= Course Logistics
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The textbook for the course is _Smith, Eggen, Andre. A Transition to Advanced
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Mathematics. 8th ed_. #smallcaps[isbn:] `978-1-285-46326-1`. Chapters 1-5 will
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be covered.
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Lecture meets every M-W-F from 12:00 -- 12:50 in Phelps 1444. Recitation meets
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M-W from 7:00 -- 7:50 in HSSB 1236.
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== Homework
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Homework is from textbook and is worth 30% of the grade, due on Gradescope.
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Homework is due every W at 11:59 PM. LaTeX is recommended for typesetting but
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of course we will be using Typst, the superior typesetting software for
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mathematics.
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Section and problem numbers should be clearly labeled and problems should be
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done on a single column.
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The lowest homework score will be dropped.
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== Exams
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Each exam is 20% of the grade. The final exam wil replace the lowest of the
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first two exam scores if it is higher.
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= Chapter 1: Logic and Proofs
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== Trivial preliminaries
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Definitions barely worth considering. Included purely for posterity.
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#definition("Proposition")[
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A proposition is a sentence which is either true or false.
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]
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#example("Primes")[
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The numbers 5 and 7 are prime.
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]
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#example("Not a proposition")[
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$x^2 + 6x + 8 = 0$
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]
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Propositions may be stated in the formalism of mathematics using connectives,
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as *propositional forms*.
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#definition("Propositional forms")[
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Let $P$ and $Q$ be propositions. Then:
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+ The conjunction of $P$ and $Q$ is written $P and Q$ ($P$ and $Q$).
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+ The disjunction of $P$ and $Q$ is written $P or Q$ ($P$ or $Q$) (here "or" is the inclusive or).
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+ The negation of $P$ is written $not P$.
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]
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#definition("Tautology")[
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A propositional form for which all of its values are true. In other words, a statement which is always true.
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]
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#definition("Contradiction")[
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A propositional form for which all of its values are false. In other words, a statement which is always false.
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]
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#problem[Prove that $(P or Q) or (not P and not Q)$ is a tautology][
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Trivial, omitted.
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]
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#example[Several denials of the statement "integer $n$ is even"][
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- It is not the case that integer $n$ is even.
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- Integer $n$ is not even.
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- $n != 2m, forall m in ZZ$
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- $n = 2m + 1, exists m in ZZ$
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]
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DeMorgan's Laws tell us how to distribute logical connectives across
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parentheses.
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#theorem[DeMorgan's Laws][
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+ $not (P or Q) = not P and not Q$
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+ $not (P and Q) = not P or not Q$
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]
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#proof[
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Trivially, by completing a truth table.
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]
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