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documents/by-course/math-8/course-notes/main.typ
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@ -17,8 +17,18 @@
#outline() #outline()
= Introduction
Math 8 is an introductory course on mathematical logic and the methods of
proof. In general it will be quite trivial for anyone somewhat familiar with
competition mathematics or proofs. If you are at UCSB, I highly recommend you
take this course as soon as possible to unlock the rest of the higher
mathematics offerings here (which are much more interesting).
= Course Logistics = Course Logistics
Everything in this section is information only valid for the Winter 2025 quarter with Professor Porter.
The textbook for the course is _Smith, Eggen, Andre. A Transition to Advanced 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 Mathematics. 8th ed_. #smallcaps[isbn:] `978-1-285-46326-1`. Chapters 1-5 will
be covered. be covered.
@ -315,13 +325,15 @@ $ P => Q and Q => P $
then showing that this fact leads to a contradiction. then showing that this fact leads to a contradiction.
A proof by contradiction of $P => Q$ proceeds by assuming $P and not Q$, then A proof by contradiction of $P => Q$ proceeds by assuming $P and not Q$, then
showing a contradiction, implying that $P$ indeed implies $Q$. showing a contradiction, forcing that $P$ indeed implies $Q$.
] ]
#definition[ #definition[
A real number $x in RR$ is called rational iff A real number $x in RR$ is called rational iff
$ exists p,q in ZZ, x = p / q $ $ exists p,q in ZZ, x = p / q $
]
#definition[
$x$ is irrational if it is not rational. $x$ is irrational if it is not rational.
] ]