Questions tagged [set-theory]
For questions about set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies, etc.
26 questions
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chaining implications and equivalences in proofs
I have a question (more of a "sanity check") regarding the interpretation of proofs one often encounters, particularly in university-level courses where proofs are written on blackboards.
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2
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Correct notation of a Sample space
From the very beginning I have used the notation $ S = \{ H , T \} $ as the Sample space for tossing a coin once and $ S = \{ HH , HT , TH , TT \} $ in the case of tossing a coin twice.I have several ...
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Whole numbers as sets vs abstracted properties of sets
I recently landed on a book written for elementary school teachers which introduced the concept of whole numbers in the following manner:
We have a set $\{\alpha, \beta, \gamma\}$. There are other ...
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Can this be a better way of defining subsets?
I remember my high school days where subsets were defined in the following manner:
Given two sets A and B, if every element of B is an element of A, then B is called a subset of A.
A common ...
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What is an example of something you might see outside of math class and how would you model that thing as a set?
In mathematics, we have sets, such as $\begin{Bmatrix}1, 2, 3 \end{Bmatrix}$ or the real-numbers, usually denoted as $\mathbb{R}$.
When teaching students about sets for the first time, it can ...
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Is it considered a mistake to use different correct notation for writing intervals?
Standard definition of writing interval states that it should be written (a,b) where a<b
Due to this being arbitrary and just a convention that we all use, would it be considered a mistake to write ...
- 21
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Mnemonics to correlate the definition of "asymmetric relation" and "antisymmetric relation" with the terms [closed]
The definitions from Kenneth Rosen textbook are as :
A relation $R$ on a set $A$ such that for all $a,b ∈ A$ ,if $(a,b) ∈ R$ and $(b,a) ∈ R$,then $a=b$ is called antisymmetric.
A relation $R$ on a ...
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Workbooks for advanced high school math topics
I'm looking for advanced workbooks and exercises for working in class (math high school/undergraduate level) covering the following topics (or some of them):
Logic and sets (propositional calculus, ...
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Fun set theory for kids
Are there some fun results in set theory to set as landmarks while introducing to kids?
For example, while introducing graph theory to kids, I could explain isomorphism via a pentagon and pentagram, ...
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Where can I find the partial order relation of prerequisites of undergraduate courses in the United States?
Let $A$ be the set of all undergraduate mathematical courses in the US and define a binary relation $\leq$ on $A$ such that for elements $a,b\in A$ (that is, $a$, $b$ are undergraduate mathematical ...
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Category mistakes regarding symbols and their impact on math (mis) understanding. ( Object symbol/ sentence symbol confusion)
A friend of mine that teaches math has made many times the following experiment :
drawing two circles on the blackboard representing two sets A and B such that A and B are disjoint
writing on the ...
user12295
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What is the best way of introducing set theory? [closed]
The students are aware of mathematical logic and proof but have not come across any of the notions of a set. What is the most natural and motivating way to introduce set theory?
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When self teaching, should I learn set theory before continuing ap calculus?
I am studying ap calculus now, before I move onto differential equations etc., but the thing I am unsure of is, should I learn set theory before continuing on my ap calculus sections?
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Why is set theory not taught at the outset of math education?
A beginner in math, reading Badiou, I found the following quote on set theory in Being and Event:
The axiomatization consists in fixing the usage of the relation of belonging, $\in$, to which the ...
Ken
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Cognitive traps in very early set theory
As of last year, I began teaching a course in theoretical computer science, and one of the topics that we cover is sets. In particular, we go over:
the actions $\cup$, $\cap$, and $-$,
...
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How can I convince my brightest student of Cantor's theory?
At the end of the mathematical high-school education I usually introduce the easiest facts of set theory: counbtability and Cantor's proof as the basis of modern mathematics. Now my brightest student ...
user8455
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Determining sets to show sufficiency of a condition?
$p \to q$ that means (among others)
$p$ is a sufficient condition for $q$.
To show the sufficiency, I teach my study by determining the set for $p$, the set for $q$ first and comparing their ...
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What to teach in Set Theory & Logic Course
I will be teaching a third-year introductory course on Set Theory and Logic soon and was hoping to get advice from this community.
I would rate my students' proof abilities as weak and was hoping to ...
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Cartesian product set
I'm preparing a task on the Cartesian product of two sets and I have run into the following confusion:
I understand that the Cartesian product is not a commutative operation. Generally speaking, AxB ...
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On using different notations for the same objects
Historically, in set theory we use two different notations to refer set theoretically same objects $\aleph_{\alpha}$ and $\omega_{\alpha}$. The folklore justification of this dual notation is that we ...
user230
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What caused the (relatively) recent popularity of set theory?
When I was growing up during the 1960s, "set builder notation" constituted a large part of what was then the "new math."
Question: When and why did "set theory" become popular in math education? ...
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Examples of cultural limitations on math education
Based on Maggie Koerth-Baker's article, "What do Christian fundamentalists have against set theory?", it seems there are some parts of culture which put some restrictions on math education.
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user230
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What is a number?
In a set theoretic point of view all mathematical objects are sets. We "call" some of them as numbers (e.g. sets in $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $Ord$, $Card$) but what is ...
user230
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How to motivate equivalence classes
Equivalence classs are very useful in mathematics, but many of the applications require further background, like quotient spaces in topology or quotient groups in algebra. One good example is residue ...
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What do you say to students who want to apply Banach-Tarski theorem in practice?
Once when I was talking about Banach-Traski theorem (paradox) I said:
OK! This is Banach-Tarski's theorem which is against our intuition but provable from our intuitive axioms! It says you can ...
user230
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How can I familiarize elementary school students with infinities larger than $\aleph_0$?
Cantor's discovery of the existence of more than one infinity was a revolutionary change in human knowledge. He defined the notion of counting by bijections and showed that one can use infinities as ...
user230