159 lines
4.6 KiB
Text
159 lines
4.6 KiB
Text
#import "./dvd.typ": *
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#import "@preview/ctheorems:1.1.3": *
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#show: dvdtyp.with(
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title: "PSTAT120A Course Notes",
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author: "Youwen Wu",
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date: "Winter 2025",
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subtitle: "Taught by Brian Wainwright",
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)
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#outline()
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= Lecture 1
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== Preliminaries
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#definition[
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Statistics is the science dealing with the collection, summarization,
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analysis, and interpretation of data.
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]
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== Set theory for dummies
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A terse introduction to elementary naive set theory and the basic operations
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upon them.
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#remark[
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Keep in mind that without $cal(Z F C)$ or another model of set theory that
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resolves fundamental issues, our set theory is subject to paradoxes like
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Russell's. Whoops, the universe doesn't exist.
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]
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#definition[
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A *Set* is a collection of elements.
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]
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#example[Examples of sets][
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+ Trivial set: ${1}$
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+ Empty set: $emptyset$
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+ $A = {a,b,c}$
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]
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We can construct sets using set-builder notation (also sometimes called set
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comprehension).
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$ {"expression with" x | "conditions on" x} $
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#example("Set builder notation")[
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+ The set of all even integers: ${2n | n in ZZ}$
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+ The set of all perfect squares in $RR$: ${x^2 | x in NN}$
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]
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We also have notation for working with sets:
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With arbitrary sets $A$, $B$:
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+ $a in A$ ($a$ is a member of the set $A$)
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+ $a in.not A$ ($a$ is not a member of the set $A$)
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+ $A subset.eq B$ (Set theory: $A$ is a subset of $B$) (Stats: $A$ is a sample space in $B$)
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+ $A subset B$ (Proper subset: $A != B$)
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+ $A^c$ or $A'$ (read "complement of $A$", and gives all the elements in the universal set not in $A$)
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+ $A union B$ (Union of $A$ and $B$. Gives a set with both the elements of $A$ and $B$)
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+ $A sect B$ (Intersection of $A$ and $B$. Gives a set consisting of the elements in *both* $A$ and $B$)
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+ $A \\ B$ (Set difference. The set of all elements of $A$ that are not also in $B$)
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+ $A times B$ (Cartesian product. Ordered pairs of $(a,b)$ $forall a in A$, $forall b in B$)
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We can also write a few of these operations precisely as set comprehensions.
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+ $A subset B => A = {a | a in B, forall a in A}$
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+ $A union B = {x | x in A or x in B}$ (here $or$ is the logical OR)
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+ $A sect B = {x | x in A and x in B}$ (here $and$ is the logical AND)
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+ $A \\ B = {a | a in A and a in.not B}$
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+ $A times B = {(a,b) | forall a in A, forall b in B}$
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+ $A' = A sect Omega$, where $Omega$ is the _universal set_.
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#definition[
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The universal set $Omega$ is the set of all objects in a given set
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theoretical universe.
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]
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Take a moment and convince yourself that these definitions are equivalent to
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the previous ones.
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#example[The real plane][
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The real plane $RR^2$ can be defined as a Cartesian product of $RR$ with
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itself.
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$ RR^2 = RR times RR $
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]
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Check your intuition that this makes sense. Why do you think $RR^n$ was chosen
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as the notation for $n$ dimensional spaces in $RR$?
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#definition[Disjoint sets][
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If $A sect B$ = $emptyset$, then we say that $A$ and $B$ are *disjoint*.
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]
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#fact[
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For any sets $A$ and $B$, we have DeMorgan's Laws:
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+ $(A union B)' = A' sect B'$
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+ $(A sect B)' = A' union B'$
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]
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#fact[Generalized DeMorgan's][
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+ $(union_i A_i)' = sect_i A_i '$
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+ $(sect_i A_i)' = union_i A_i '$
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]
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== Sizes of infinity
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#definition[
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Let $N(A)$ be the number of elements in $A$. $N(A)$ is called the _cardinality_ of $A$.
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]
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Sets are either finite or infinite. Finite sets have a fixed finite cardinality.
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Infinite sets can be either _countably infinite_ or _uncountably infinite_.
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When a set is countably infinite, its cardinality is $aleph_0$ (here $aleph$ is
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the Hebrew letter aleph and read "aleph null").
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When a set is uncountably infinite, its cardinality is greater than $aleph_0$.
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#example("Countable sets")[
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+ The natural numbers $NN$.
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+ The rationals $QQ$.
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+ The natural numbers $ZZ$.
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+ The set of all logical tautologies.
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]
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#example("Uncountable sets")[
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+ The real numbers $RR$.
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+ The real numbers in the interval $[0,1]$.
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+ The _power set_ of $ZZ$, which is the set of all subsets of $ZZ$.
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]
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#remark[
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All the uncountable sets above have cardinality $2^(aleph_0)$ or $aleph_1$ or
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$frak(c)$ or $beth_1$. This is the _cardinality of the continuum_, also
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called "aleph 1" or "beth 1".
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However, in general uncountably infinite sets do not have the same
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cardinality.
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]
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#fact[
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If a set is countably infinite, then it has a bijection with $ZZ$. This means
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every set with cardinality $aleph_0$ has a bijection to $ZZ$. More generally,
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any sets with the same cardinality have a bijection between them.
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]
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This gives us the following equivalent statement:
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#fact[
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Two sets have the same cardinality if and only if there exists a bijective
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function between them. In symbols,
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$ N(A) = N(B) <==> exists F : A <-> B $
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]
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