281 lines
7.3 KiB
Text
281 lines
7.3 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 2025",
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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 will replace the lowest of the
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first two exam scores if it is higher.
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= Meeting #datetime(year: 2025, month: 1, day: 6).display()
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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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#fact[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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Also, propositional forms obey commutative, associative, distributive laws,
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which can be trivially obtained from symbolic manipulations and will not be
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restated. Together with the double negation law and the _law of the excluded
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middle_, these comprise the axioms of a system of propositional logic.
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#fact[
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We abbreviate propositional forms by eliding parentheses, according to the rules:
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+ $not$ is applied to the smallest proposition following it.
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+ $and$ connects the smallest propositions surrounding it.
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+ $or$ connects the smallest propositions surrounding it.
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]
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= Notes on Logic and Proofs, 1.2
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_Prototypical example for this section:_ If $sin pi = 1$, then $6$ is prime.
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#definition[
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For a *antedecent* $P$ and *consequent* $Q$, the *conditional sentence* $P =>
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Q$ is the proposition "If $P$, then $Q$."
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]
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#remark[
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The statement $P => Q$ states $P$ _implies_ $Q$ and is only false if $P$ is
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true and $Q$ is false, since this is the only case where $P$ did not imply $Q$.
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]
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A conditional may be true even when the antedecent and consequent are unrelated.
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= Lecture #datetime(day: 8, month: 1, year: 2025).display()
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== More propositional forms
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#definition[
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Let $P$ and $Q$ be propositions. The *biconditional sentence*
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$ P <=> Q $
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is true exactly when $P$ and $Q$ are both true or both false.
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]
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#example[Ways of stating $P <=> Q$][
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- $P$ if and only if $Q$
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- $P$ iff. $Q$
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- $P$ is equivalent to $Q$
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]
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#exercise[
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Translate each statement into symbols, where $a$ is a fixed real number.
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+ $a > 5$ is sufficient for $a > 3$
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+ $a > 3$ is necessary for $a > 5$
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+ $a > 5$ only if $a > 3$
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+ $|a| = -a$ whenever $a < 0$
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+ $|a| = 2$ is necessary and sufficient for $a^2 = 4$
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]
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#definition[
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#set enum(numbering: "a.")
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Let $P$ and $Q$ be propositions.
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+ The converse of $P => Q$ is $Q => P$
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+ The contrapositive of $P => Q$ is $not Q => not P$
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]
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#theorem[
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Let $P$ and $Q$ be propositions. Then:
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]
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#example[
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If $f(x)$ is differentiable at $x = a$, then $f(x)$ is continuous at $x = a$.
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$ P => Q $
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+ $not Q => not P$: if $f$ is not continuous at $x=a$, then $f$ is not differentiable at $x = a$.
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+ $Q => P$: if $f$ is continuous at $x = a$, then $f$ is diffferentiable at $x = a$.
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]
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#fact[
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We apply our new logical connectives in the following order: $not, and, or, => , <=>$
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]
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#example[
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Include parentheses to clarify the expression.
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$ P or Q => not R <=> S and T $
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]
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#theorem[
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#set enum(numbering: "a.")
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For propositions $P$, $Q$, and $R$, the following are equivalent:
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+ $P => Q "and" not P or Q$
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+ $P <=> Q "and" (P => Q) and (Q => P)$
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+ $not (P => Q) "and" P and not Q$
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]
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== Quantified statements
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#definition[
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A *predicate* or *open sentence* is a sentence involving one or more variables.
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]
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#example[
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Consider the open sentence $P(x,y): x^2 + y^2 = 25$. Write a true and a false proposition.
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$
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P(3,-4) &: 3^2 + (-4)^2 = 25 &"(true)" \
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P(2,0) &: 2^2 + 0^2 = 25 &"(false)" \
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$
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]
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#definition[
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The *universe* is the set of all objects available for substitution into an open sentence. Denoted $U$.
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]
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#definition[
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A *truth set* is all objects in $U$ that make an open sentence true.
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]
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#example[
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Let the universe be the set of all real numbers for the open sentence $P(x) : x^2 + x = 6$. Find the truth set.
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$
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U = RR \
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"truth set:" {2,-3}
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$
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]
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#definition[
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Let $P(x)$ be an open sentence with variable $x$.
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The *universal quantifier* is the sentence
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$ forall x in U, P(x) $
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The *existential quantifier* is the sentence:
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$ exists x in U, P(x) $
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The *unique existence quantifier* is the sentence
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$ exists! x in U, P(x) $
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]
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#example[
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Let the universe be the set of all real numbers and consider the open sentence
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$ P(x) : x^2 + 1 >= 0 $
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Consider the quantified sentence $forall x in U,P(x)$. Then
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$ forall x in RR, x^2 + 1 >= 0 $
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is a true statement.
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However, if instead $U = CC$, then the sentence is false.
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]
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#example[
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Let the universe be the set of all real numbers and consider the open sentence
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$
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Q(x) "where" x in ZZ \
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R(x) "is a perfect square" \
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$
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Consider the quantified sentence
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$ exists Q(x), R(x) $
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]
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#example[
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Let the universe be the set of all real numbers and consider the open sentence
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$ P(x,y) : y = x^3 + 4 $
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Consider the quantified sentence
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$ forall y in U, exists! x in U, P(x,y) $
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It is true because $P(x,y)$ is injective (one-to-one).
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]
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