# 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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### Showing $R$ is transitive and reflexive $\to$ $R=R^2$, $R$ is transitive and reflexive $\to$ $R=R^2$

Let $R$ be a relation over $A$. Define $R^{-1}, R^2$ like so: $aR^{-1}b \iff bRa\\ aR^2b\iff\exists _{c\in A}(aRc\wedge cRb)$ Prove: $R$ is transitive $\iff$ $R^2\subseteq R$ $R$ ...
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### Show $\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$

$\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$ Sorry if this question has been asked already but I couldn't find it on this site. I assume by definition of a ...
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### Unique expression as disjoint union of indecomposable subsets

Let $f:A \to A$ be a function, we say that $B \subseteq A$ is $f$-invariant iff $f(B) \subseteq B$. We say that an invariant subset is indecomposable iff it cannot be expressed as a union of non-...
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### $\{h\in A^B|h \text{ is invertible}\}$ is equiumerous to $\{k\in B^A|k \text{ is invertible}\}$ and $\aleph_0$ right invertibles for a function

1.Let $A,B$ be sets, prove: $\{h\in A^B|h \text{ is invertible}\}$ is equinumerous to $\{k\in B^A|k \text{ is invertible}\}$ 2.Let $A,B$ be sets and a function $f\in A^B$ give an example right ...
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### If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?

If the empty set is a subset of every set, why it isn't written with the elements of a set? like so $\{1,2,3,\emptyset\}$ Or why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$? I know one has two elements ...
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### Expressing $\{\mathbf x: x_n=x_{n+1}\text{ for every$n$prime number}\}\subseteq R^\omega$ as cross product of subsets of $\mathbb R$

The question is: can we express such a set in terms of cross products between subsets of $\mathbb R$? We would have $\{\mathbf x=(x_1,x_4,x_4,x_4,x_6,x_6,x_8,x_8,x_9,x_{10},x_{12},x_{12},\dots)\}$. ...
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### Well ordering of $\mathbb{N}$ using inductive sets

In this book (Elementary Real Analysis by Thomson-Bruckner p.22), $\mathbb{N}=\left\{ 1,2,...\right\}$ (In some, $0\in\mathbb{N}$). In an exercise, a set $S\subset\mathbb{R}$ is inductive if $1\in S$ ...
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### Give an example of a set which is not transitive

Transitive set: set $x$ is transitive if $\forall y\in x(y\subseteq x)$ I think $\{\varnothing\}$ is not transitive since $\varnothing\in\{\varnothing\}$ but $\varnothing\not\subseteq\{\varnothing\}$ ...
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### Proof: There cannot be a universal set / set of all sets [duplicate]

I am new to mathematics as well as to math.stackexchange so my question will be a very basic one and I might have related questions that will seem very basic if not trivial. My first question is the ...
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### Difference between a criteria of well-ordered and transitive in terms of an ordinal?

From what I gather a set $x$ is $transitive$ if whenever $y \in z , z \in x \rightarrow y \in x$ And one of the properties of a well-ordered set is that $x \in y , y \in z \rightarrow x \in z$ Now ...
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### Nullary Arithmetic Product (at Wiki)

In Nullary Arithmetic Product at Wiki, we are given a sequence of numbers $a_1, a_2, a_3\ldots$ The product of the first $m$ elements of this sequence is given there by $P_m=a_m \cdot P_{m-1}$ ...
Just wanted to make sure the way I approach this was correct because it seemed a bit too simple for an answer: Question: Let $\mathcal P \left({X}\right)$ represent the power set of $X$. Let $f: X \... 2answers 96 views ### Demonstrate currying via homomorphism You can demonstrate currying for two-argument functions is possible showing there's a isomorphism between$(A^B)^C \cong A^{B \times C}$. That is, the set of functions$(C \rightarrow B) \rightarrow A$... 2answers 113 views ### confusion over Finite intersection property It is stated that$A_n={(\frac{-1}{n},\frac{1}{n})}$, then arbitrary intersection of open sets need not be open is true as in this case$\bigcap_{i=1}^{\infty}=\left \{0 \right \}$is not open. Now ... 1answer 49 views ### Is this a topological closure operation? Does any relation$\propto\,\subseteq X\times \mathcal P(X)$that extends$'\!\!\in'$in the way that:$x\in M\Rightarrow x\propto M\neg\exists x\in X:x\propto\emptysetx\propto A \subseteq B ...
I have the following problem: Let $B_n = [0, 1 + n^{−1} ),\space\space T_n = [0, 1 − n^{−1} ],\space\space n \in \mathbb{N}$. Show that $\bigcap B_n = [0, 1]$ and $\bigcup T_n = [0, 1)$ ...