# Tagged Questions

This tag is for elementary questions on set theory, spanning topics usually found in introductory courses in set theory, in addition to review sections of graduate textbooks in the same field. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, (un)...

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### Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and calculus....
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### Cardinality of set of groups

After analyzing this question I started wondering. Thoughts. Everyone can give a simple example of a countable group $G, |G| = \aleph_0$ which has uncountable number $2^{\aleph_0}$ subgroups, for ...
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### Let $A$ be a set with $m$ elements and let $B$ be a set with $n$ elements where $m,n\in \omega$ and $m>n$. If $f:A\to B$, then $f$ is not injective

So I am still learning how to work with infinite sets, and this particular problem is giving me some issues. Right now, I am trying to pick some $x_1,x_2\in A$ such that $f(x_1) = f(x_2)$ to serve as ...
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### Definition of Ordered Pair

This is probably a more open ended question. Kuratowski's definition of ordered pair is that $(a,b) = \{\{a\},\{a,b\}\}$. This is basically just subset of a power set. But say we have a probability ...
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### Some Trouble Understanding set theory

I'm currently in a discrete mathematics class and we've recently been discussing set theory. I feel like I have basic understanding of how to actually prove set relations when a question asks to do so....
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### Bijective continuous map between $\mathbb{R}$ and $\mathbb{R}^2$

I am currently attending a course on point-set topology and fundamental group. Today we proved in class that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic with their normal Hausdorff topologies....
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### The supremum of any set of cardinals (considered as a set of ordinals) is again a cardinal.

An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ ...
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