Real Analysis Course #1 - Ordered Sets

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Here's the first video in a series of many on the topic of mathematical real analysis. This course is fundamental and usually required for all math majors. Let's dive in with ordered sets and the study of Real Analysis!

**Note** that in definition i) exactly 1 and only 1 of the three cases must be true!

#realanalysis #brithemathguy #math

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.
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🎓Become a Math Master With My Intro To Proofs Course! (FREE ON YOUTUBE)

BriTheMathGuy
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This playlist is really needed, the only other real analysis stuff on YT are university lectures which are way too slow

ClickPhil
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This a really some of the core backbones of Mathematics studied during my undergraduate classes. I love abstract Mathematics and your explanation resonate this feeling today!

aljebraschool
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I have my first real analysis course next year, so I'm looking foreard to getting a bit ahead with these videos!

qaysdaou
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I just bought a new real analysis textbook, and this looks great to follow along with it!

cconn
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Thank you!! Taking real this semester and definitely needed a different source of instruction

Jack-ecii
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I've always wanted to learn real analysis! Thank you very much!!!

hongjooryoo
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You are trying to define a (strict) total/linear order on a set. Since you presented a strict order, you left out the irreflexive property. A strict partial order is irreflexive (i.e ~(x<x), where ~ means negation), transative(your (ii)) and asymmetric(i.e. if x<y then ~(y<x)) which need not be stated as it is an implication of the irreflexivity and transativity. A total order is a partial order that is connexive(your (i)), i.e. every element is comparable to every other element - a chain. It can be more convenient for you to define a non-strict total relation using your (i) -connexity, (ii) transativity would stay but then the relation would now be reflexive and you would need to add antisymmetry(i.e. x<=y & y<=x imply x=y), as a non-strict partial order is a binary relation on a set that is irreflexive, antisymmetric and transative. Then define x<y as the abbreviation of x<=y & ~(x=y). What's important in real analysis is that the reals are a complete ordered field, and in fact (up to isomorphism) the set of real numbers is the only such one. The completeness property being equivalent to the Nested Interval theorem, Dedekind Completeness, Monotone convergence, Bolonzo-Weierstrass, Intermediate value theorem etc.The rationals are (up to isomorphism) the smallest ordered field. Also a well-order on a set is a binary relation on a set such that every non-empty subset has a least element in that ordering, it is trival to see every well-order is a total order and an equivalent of the axiom of choice is that every set can be well-ordered, that is there exists a relation (a subset of the Cartesian product of the set with itself) that is a well-order.

woodin_cardinal-
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Just what I needed! subscribed and liked. Keep it up.

detectivetacox
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Literally thank you so much for posting this, l've really wanted to get into this subject but it seemed too intimidating to me :/. l'm really really looking forward to more of your videos, please keep it up <3

Anna-jycj
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My adhd thanks you deeply I could focus on my lectures at all

ericaporter
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Are you going to construct the set of numbers through peano axioms (N), equivalence classes (Z and Q), and dedekind / cauchy sequences (R)?

breisfm
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well this is a bit late because i just took my midterm on sequences limits and continuity hahahha

raichuk
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Coming back to this ten years after I took real analysis junior year of college. Question - does proving that x and y are ordered imply that all elements of S are ordered? Is that specified by the "for all x, y" condition?

commirevo
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Does that mean, all sets are ordered? I can't think of any counter examples.

zarifmuhtasim
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How could a set not be ordered at all?

georgigachev
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Can you make the same kind of videos for Abstract algebra?

factsss
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I'm still waiting for the crash course on subsequences smhhh :c great stuff btwww

milopc
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what is the ratio of like to dislike in this video... hah undefined 156 : 0 = 156/0

evenprime