# 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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### Understanding the solutions to questions concerning cardinalities and power sets.

Let $A = \{1, 2, 3, ... , n\}$. Find the cardinalities of the following sets: $\{(a, S) \mid a \in S, S \in P(A)\}$ $\{(S, T) \mid S \in P(A), T \in P(A), S\cap T = \emptyset \}$ ...
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### Let $A = \{1, 3, 5, 7, 9\}$ and $B=\{3, 6, 9\}$. Find each of the following: (i) $A \cup B$ (ii) $A \cap B$ (iii) $A − B$ [closed]

Let $A = \{1, 3, 5, 7, 9\}$ and $B=\{3, 6, 9\}$. Find each of the following: (i) $A \cup B$ (ii) $A \cap B$ (iii) $A − B$ I am doing a test today which I must prove this kind but honestly ...
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### Trouble with a proof exercise in Set Theory regarding subsets and Power Sets

Question as posed: Let U be any set. Prove that for every $A\in\mathcal{P}(U)$ there is a unique $B\in\mathcal{P}(U)$ such that for every $C\in\mathcal{P}(U)$, $C\setminus A=C \cap B$. Proof (so far)...
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### Hamel Bases: Cardinality? [duplicate]

Every vector space admits a Hamel basis by AC. That is there are maximally linear independent sets. But how to prove their cardinalities necessarily agree? ..I couldn't really find any reference.
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### How to show that the following function is bijective?

If we have the function $c : \mathbb{N}^2 \rightarrow \mathbb{N} : (x,y) \rightarrow 2^x \cdot (2y+1) -1$ how to show that this function is bijective? So I thought the easiest way is to show that is ...
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### An example of an infinite set with $S$ with there exists some cardinality between $S$ and $P(S)$.

I just read about Continuum Hypothesis which states that there is no set $S$ with the cardinality of $S$ is strictly larger than $\mathbb{N}$ and strictly smaller than $\mathbb{R}$. I recall that in ...
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### Geometric interpretation of Cantor's infinite list of real numbers

Cantor imagined the list of real numbers and demonstrated by diagonalization that the list can never be complete, but I wonder if such a diagonalization is even possible. If we try to build his list ...
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### Is $f \circ g$ invertible in the diagram below?

I was working through Can the composition of two non-invertible functions be invertible? For the image below is $f \circ g$ invertible? Thanks!
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### Partition of complete boolean algebra induces partition on elements

Given a complete boolean algebra B, and two partitions W and T of B, why is it true that W induces a partition on every element of T? (And is this true more generally - does W induce a partition on ...
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### If $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$

I've been thinking about the following claim: Let $A$ be a set and $|A|$ his cardinality. For every cardinal $\lambda$ with $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$. ...
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### There exist partition of set $X$ due to relation $R$ and surjection $g: X\to X|_R$ and injection $h:X|_R \to Y$ such as $h \circ g=f$

$f: X\to Y$ is function. Prove: There exist partition of set $X$ due to relation $R$ on $X$ and surjection $g: X\to X|_R$ and injection $h:X|_R \to Y$ such as $h \circ g=f$
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### Proof of a Surjective Function

I've run into a question in my textbook and I'm not sure if I understand fully the answer from the solution manual. Here is the question: Problem: Suppose that $f: A \rightarrow B$ is any function. ...
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### Is this a bijective function for $f:(0,1) \rightarrow (-2,5)$?

$f:(0,1) \rightarrow (-2,5)$ I'm basically trying to prove the two intervals above have the same cardinality by finding a bijective function. I'm not sure I did it properly but the function I found ...
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### Given $C \subset A \subset X$, why is that $C$ is closed in $X$ if $A$ is closed, $C$ is open in $X$ if $A$ is open?

I want to understand a result discussed here : Subspace of a normal space Let $(X, \mathfrak{T})$ be a topological space. Given $C \subset A \subset X$, let $C$ be a closed set in $A$, then claim ...
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### Find a function that is a bijection $f:(0,1) \rightarrow (1, \infty)$

Find a function that is a bijection $f:(0,1) \rightarrow (1, \infty)$ I am to assume the intervals have the same cardinality. I honestly don't even know how to begin with this. Can you provide me ...
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### Are there any constructive axioms which disprove the continuum hypothesis?

I understand that the Continuum hypothesis is independent of ZFC, so that we may comfortably add either the continuum hypothesis or its negation to ZFC without creating any paradoxes (unless ZFC had ...
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### How to prove the power set of the rationals is uncountable?

Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
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### My proof that $f^{-1}(D_1 \cap D_2) = f^{-1}(D_1) \cap f^{-1}(D_2)$

I'm currently self studying proof and set-theory, and I'm quite new to both of them. As an exercise, I'm practicing proving some basic theorems, so it'll be great if you can give me some feedback on ...
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### When is it true that $V \subseteq \overline V \subseteq U$ will hold for open sets?

Let $(X, \mathfrak{T})$ be a topological space Let $V, U \in \mathfrak{T}$, and suppose that $V \subseteq U$ Then when it is true that $V \subseteq \overline V \subseteq U$, where $\overline V$ is ...
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### How do I represent these complements in membership tables?

I couldn't find examples of some of these elsewhere, so I'm just kind of guessing here. I'm fairly certain I've made a mistake somewhere, particularly the columns with complements. I've placed ...
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### Can someone tell me how to prove $\cap_{i\in I}(B \setminus A_i) = B \setminus (\cup_{i\in I} A_i)$ [closed]

Can someone tell me how to prove $\cap_{i\in I}(B \setminus A_i) = B \setminus (\cup_{i\in I} A_i)$
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### Let $\sigma :\{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}$ be permutations such that $\sigma^{-1}(j) \leq \sigma(j)~\forall j, 1 \leq j \leq 5$.

Which of the following are true? $\sigma \circ \sigma(j)=j~\forall j, 1 \leq j \leq 5$. $\sigma^{-1}(j)=\sigma(j)~\forall j, 1 \leq j \leq 5$. The set $\{k: \sigma(k) \neq k \}$ has even number of ...
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### In set theory, to prove $A=B$, is it always necessary to prove that $A\subset B$ and $B\subset A$

I am sorry if this a stupid question. I am currently studying set theory, and one thing that is really annoying me is ways of proving two sets equal. In each and every theorem I have come across, my ...
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### Under what condition does $f(f^{-1}(f(A))) = f(A)$?

Basic question regarding function. Let $f: X \to Y$, then for what $f$ does $f(f^{-1}(f(A))) = f(A)$? hold? Obviously this relationship holds when $f$ is a bijection. This does not hold when $f$ ...