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How to use the element method to prove the following sets are equal?

I have been asked to describe the following sets, and then prove my answers using the element method, but i am not sure how to do this. I am trying to prove that (b) is equal to $0$ as $i$ approaches...

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Let $(X, mathfrak T)$ be a topological space and supposed that A is a subset...

Let $(X, mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) cap Cl(X-A)$. I know this is a true statement. I am trying to prove if because I would also like...

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Existence of infinite set and axiom schema of replacement imply axiom of...

I’m self-teaching an intro to set theory course, and came across this exercise: Show that the existence of an infinite set is equivalent to the existence of an inductive set. For the notion of...

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Is the Cartesian product of two uncountable sets uncountable? [duplicate]

This question already has an answer here: Is the set of all pairs of real numbers uncountable? 2 answers

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Set Theory – Simplify expression

Can the following be simplified? It’s been a long time since I did set theory and I don’t remember my simplification rules. This is probably totally easy… can I simplify this any further? $(A cap B)...

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Why does $bigcap_{m = 1}^infty ( bigcup_{n = m}^infty A_n)$ mean limsup of...

Why does $bigcap_{m = 1}^infty ( bigcup_{n = m}^infty A_n)$ mean the limit superior of sequence of set? I’m not getting it. ${A_n}$ is a sequence of set in $S$. I do know what limsup means for a...

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Can Zorn's Lemma be 'inverted' like this:?

Let $R$ be a (commutative) ring not equal to $0$. I want to show that the set of prime ideals of $R$ has a minimal element w.r.t. inclusion. This may be a wholeheartedly wrong attempt, but I thought...

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Axiom of choice? or other theorems? [duplicate]

This question already has an answer here: Prove that every set with more than one element has a permutation without fixed points 5 answers

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Prove that $mu(x,z)geq min left { mu(x,y),mu(y,z) right }$ for $x,y,zin {...

Let $x,y,zin M:={ 0,1}^{mathbb{N}}$ and define $mu(x,y)=min{ nin mathbb{N}mid x_{n}neq y_{n}}$. I want to show that $mu(x,z)geq min left { mu(x,y),mu(y,z) right }$. I have tested that it is true if I,...

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If I use an element chasing proof, can I prove the identity of two sets false...

For example, if A=(B∖C) is true, the following are true: ∀x(x∈A→x∈B∧~x∈C) ∀x(x∈B∧~x∈C→x∈A) (A=(B∖C) is just an arbitrary example; I don’t really know its truth value) But this sounds really just like a...

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What is the cofinality of $2^{aleph_omega}$

There is a similar question in this site but I am not satisfied with the answer, which is basically the same as the proof in the mentioned textbook. The book(Karel Hrbacek&Thomas Jech, Introduction...

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Why is this not a proof of Schroeder-Bernstein?

We can show that if $f: A rightarrow B$ is injective then $|A| leq |B|$ and if $g: B rightarrow A$ is injective then $|B| leq |A|$ so $|A| = |B|$. By the definition of having equal cardinality, there...

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Proof that every field $F$ has an algebraic closure $bar F$

I am reading the book A First Course in Abstract Algebra written by Fraleigh and I do not really understand the proof of theorem 31.22, that every field $F$ has and algebraic closure $bar F$. I notice...

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Question about set of all functions and the power set of a set

Hi so we know the set of all functions from a set X $phi rightarrow$ {0,1} create a one to one correspondence from the power set of X to the set of all functions but we are looking at certain subsets...

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Can I prove set propositions using first-order logic?

I’m studying logic and sets and I have to say there’s a strong similarity between the two. Most boolean/logic axioms also apply to sets. At the end of my course I also studied first-order logic (or...

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If A and B are disjoint and B and C are disjoint so $Acup C$ and B are disjoint

Prove: If A and B are disjoint and B and C are disjoint so $Acup C$ and B are disjoint We know that $Acap B=emptyset wedge Bcap C=emptyset rightarrow (Acap B)cap (Bcap C)= emptyset rightarrow (Acap...

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What's the meaning of an element that belongs to the same element?

In classical set theory, if I consider that $x$ is an element, which means it is not a set, can I write $x in x$ ? If yes, what this would mean? Correct me if I am wrong, but I don’t need to have some...

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Chain of length $2^{aleph_0}$in $ (P(mathbb{N}),subseteq)$

How can I find a chain of length $2^{aleph_0}$ in $ (P(mathbb{N}), subseteq )$. The only chain I have in mind is $${{0 },{0,1 },{0,1,2 },{ 0,1,2,3},…,{mathbb{N} } }$$ But the chain is of length...

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Are there more transcendental numbers or irrational numbers that are not...

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don’t know where to even start with this one whatsoever. Any help? Or is it unsolvable?

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order of infinite countable ordinal numbers

I’m trying to understand ordinal arithmetic. If one had an ordered list of the some subset of countable ordinal numbers, what order would the following 6 countably infinite ordinals be in? If the...

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