# 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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### Injective map from one set to other

In the theorem of Schroder-Bernstein, it is assumed that, given two sets $A$ and $B$, and there is an injective map from $A$ to $B$ and an injective map from $B$ to $A$. It then concludes that there ...
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### entry vs. component

When speaking about tuples, I used to say and write "The number $a$ is the first component of the tuple $(a,b,c)$, and $c$ is its last component". This all went well until I had to speak about ...
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### Is this an open or closed set?

$S=\{5+\frac{(-1)^n}{n} \; : \; n \in \mathbb N\}$ According to my calculations this set has a lower bound of $4$ and an upperbound of $5$; however, since $4$ is reachable by the set it is a minimum ...
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### Counting subsets of different sizes of a set

Let $A$ be a non-empty set, and $\mathcal{P}^*(A)$ denote the power set of $A$ excluding empty set. There is a natural equivalence relation on $\mathcal{P}^*(A)$: for $S_1,S_2\in \mathcal{P}^*(A)$...
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### Set notation for unordered cartesian product

In the question unordered cartesian product an shorthand notation for the unordered cartesian product was discussed but without any standard notation. So my question is what would be the explicit ...
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### $R$ be infinite comuttaive ring with unity , $M,N$ be modules over $R$ , $f:M \to N$ be surjective module homomorphism ; then $|M|=|N ||ker f|$?

Let $R$ be an infinite commutative ring with unity , $M,N$ be modules over $R$ , let $f:M \to N$ be a surjective module homomorphism ; then is it true that $|M|=|N || \ker f|$ ($M,N$ are not ...
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### Why do we need axiom of choice for this?

The answerers on this question say that we need AoC (or some variant thereof) to prove that every infinite set has a countably infinite subset. In my view, choice is not needed but the answerers are ...
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### How to determine the existence of all subsets of a set?

Given The definition of subset; The axiom of power set: for any set $S$, there exists a set $\wp$ such that $X \in \wp$ if and only if $X\subseteq S$ we know what a subset is and what a power set ...
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### Ordinal Numbers: is it possible to have an isomorphic copy of an ordinal $\alpha$ inside an ordinal $\beta < \alpha$?

Could there be a function $f:\alpha \to \beta$ order-preserving and injective such that $\alpha$ and $\beta$ are ordinal numbers with $\alpha > \beta$?
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### How to write a set with an index

I'd like to write a set $\{x_1, x_2, ..., x_n\}$ in a simple way. What is a popular way? In my high school, I wrote it as $\{x_i\}_{i=1}^{n}$. Is it a correct way?
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### Simplify the following question [on hold]

Simplify the following expression: $$(A\cup B)\cap (A\cup C)$$
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### If $f: A \rightarrow B$ is surjective, and $A, B$ are nonempty sets, and $X \subseteq A$, does $f(A) - f(X) = f(A - X)$?

I'm working on a proof, and the proof will be complete if this is true... but I can't find a theorem in my book that explains whether or not this is true.
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### Question about dictionary orders for $(\mathbb{Z}_+^{\omega})$

I just want to make sure I understand the explanation stated below. According to the order relation stated below we have $(a_0,a_1,...) < (b_0,b_1,..)$ if $a_i = b_i$ for finitely many values ...
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### Subtle difference between statemetns invloving negation in set theory.

What is the difference between the statements $x$ is not in an infinite number of sets $E_n$ and ...
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### Does this thing I'm calling 'the operationalization of $x$' have an accepted name?

Given a set $X$ and an element $x \in X$, we can turn $x$ into a function denoted $\tilde{x}$ as follows: for any set $Y$ and any function $f : X \rightarrow Y$, define $$\tilde{x}(f) = f(x).$$ For ...
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### Can $\bigcap_{B\in A}B=\emptyset$ Given that all elements of $A$ are inductive sets?

I am reading a course in mathematical analysis vol 1 by J.H. Garling. He defines a successor set as one that (1) contains $\emptyset$, and (2) contains $a^+$ whenever it contains $a$ (where $a^+$ is ...
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### Proof-verification: $A\times B \subset C \times D \Rightarrow A\subset C$ and$B \subset D$.

I think I have a proof but Munkres' statement "assuming $A$ and $B$ are nonempty" is making me unsure. [EDIT to clarify: This statement is given twice, once without the "assuming nonempty" and once ...
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### Finding equivalent statements with quantifiers

Find equivalent pairs: a. $\forall x(P(x)\land Q(x))$ b. $(\forall x(P(x))\land (\forall xQ(x))$ c. $\exists x(P(x)\land Q(x))$ d. $(\exists x(P(x))\land (\exists x Q(x))$ Are ...
The topology generated by a subbasis $\mathcal{S}$ is defined as the colection $\tau$ of all unions of finite intersections of elements of $\mathcal{S}$. I want to formalize $\tau$ as something ...