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, ...

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3answers
55 views

What is this set $\mathbb{R}$ mod $2\pi$

What does it mean for S = $\mathbb{R}$ mod $2\pi$? Can someone please explain as this notation is new to me.
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1answer
42 views

Are there Elements such that this Set Relationship is true?

I have a set containing a set denoted by: $\{\{?, ?\}\}$ And am looking to list elements, should there be any, such that: $ \{\{?, ?\}\} \subseteq \{1, 2, \{3, 4\}\} $ Any help would be ...
3
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1answer
90 views

Theorems not Formulable in Set Theory

Several sites I have been reading say that set theory is a good foundation for mathematics because virtually every theorem can be cast into a theorem in set theory. What is an example of a theorem ...
0
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2answers
23 views

Show $f^{-1}(A^c)=(f^{-1}(A))^c$ [duplicate]

Let $f: X \to Y$, and $A\subseteq Y$. Show that $f^{-1}(A^c)=(f^{-1}(A))^c$ I know how to prove that $f^{-1}(A^c)\subseteq(f^{-1}(A))^c$, but stuck on proving $(f^{-1}(A))^c\subseteq f^{-1}(A^c)$. ...
0
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1answer
24 views

Are the following families of sets closed under intersection?

Problem Statement Let $X$ be any set whatsoever, and let $f:X\to X$ be any function. Note that in general, no structure is imposed on $f$ whatsoever (i.e. continuity, linearity, etc). The problem is ...
2
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1answer
51 views

Cardinality of polynomials with real coefficients

What is the cardinality of the set of all polynomials with real coefficients? I know the power set of R is "more infinite" than R, so to speak, but I'm unsure of how to prove that there does or does ...
0
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2answers
65 views

If $A\dot{-} B$ is countable and $B \dot{-} C$ is countable then $A\dot{-} C$ is countable? [closed]

Prove that: If $A\dot{-} B$ is countable and $B\dot{-} C$ is countable then $A \dot{-} C$ is countable? If not give a counter-argument
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1answer
17 views

Can you give me an example of x($X/\mathscr T$)y?

Definition 7. Let $\mathscr T$ be a partition of a nonemptyset X. We define a relation $X/\mathscr T$ on X by x($X/\mathscr T$)y if and only if there exists a set $A \in \mathscr T$ such that $x, y ...
0
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1answer
31 views

If $F$ is a one-to-one function, then if $y \in ranF$, then$ f(f^{-1}(y)) = y$

This is not the proof given in the lecture, but I found a way that seemed far more intuitive to me, so I wanted to check if it was right. $F$ is one-to-one, so there's only one $y : f(x) = y$. If $y ...
0
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1answer
24 views

Why $y/\mathscr E$ is an element of $X/\mathscr E$ when it's defined $X/\mathscr E=\{\,x/\mathscr E\mid x\in X\,\}$?

"Theorem 4 Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. Then $X/\mathscr E$ is a partition of $X$. [Proof] By Theorem 3(a) and Definition 6, $X/\mathscr E =\{\,x/\mathscr ...
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2answers
38 views

Show that $(A \cup C) \cap (B \cup C') \subset A \cup B$

I am trying to prove that $$(A \cup C) \cap (B \cup C') \subset A \cup B$$ So far I got: $$(A \cup C) \cap (B \cup C') \subseteq (A \cup C \cup B) \cap (B \cup C' \cup A)$$ $$(A \cup B \cup C) \cap ...
0
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0answers
25 views

Set, n-Tuple, Vector and Matrix — links and differences

I know this question has been asked like 1000 times, however all supplied answers were not really satisfying to me. My question concerns the similarities and differences between these mathematical ...
0
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1answer
25 views

What's the difference between $x/\mathscr E$={$y \in X$∣y$\mathscr E$x} and X/$\mathscr E$={x/$\mathscr E$∣$x\in X$}?

For example Let X={0, 1, 2, 3, 4} and $\mathscr E$ is an equivalence relation on X. $\mathscr E$ is defined as $\mathscr E$ ={(0, 0), (0, 4), (1, 1), (1, 3), (2, 2), (4, 0), (3, 3), (3, 1), (4, 4)}. ...
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3answers
72 views

Are “formulas” in Axioms of ZFC indefinite?

There is the Separation Schema in Axioms of ZFC. Where do "formulas" in this axiom comes from?Are they indefinite and is ZFC actually something like "ZFC(X)" where X is a variable which denotes a ...
1
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2answers
49 views

If two sequences of sets have the same finite unions, then the infinite unions are also the same [closed]

I am stuck in the proof of the following: If $\bigcup_{n=1}^k A_n = \bigcup_{n=1}^k B_n$ for every $k$, then $\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty B_n$. I can intuitively think this ...
1
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3answers
51 views

Dumb question, $A \subset B$, what is $A - B$

I know this is a super dumb question, but if you had a little set $A$ contained in a bigger set $B$, what the set difference $A - B$? $B - A$ is well defined and I can visualize it in my head ( donut ...
0
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3answers
41 views

Cardinality and order relations on $ \Bbb{C} $ and $ \Bbb{R} $.

It has been demonstrated that complex numbers have cardinality $ \aleph_{1} $. However, it can also be shown that the complex numbers cannot be made an ordered field. How can these two facts coincide? ...
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2answers
129 views

Can a diagonal be longer than the list being diagonalized?

If we have a list of rational numbers like this... $\ 0.n_1,_1 $ where $\ n_1,_1 $ is a digit from $\ 0$ to $\ 9$ $\ 0.n_1,_2 n_2,_2 $ $\ 0.n_1,_3 n_2,_3 n_3,_3 $ $\ 0.n_1,_4 n_2,_4 n_3,_4 ...
2
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1answer
30 views

What is the difference between disjoint union and union?

If $S = A \cup B$, then $S$ is the collection of all points in $A$ and $B$ What about $S = A \sqcup B$?, I think disjoint union is the same as union, only $A, B$ are disjoint. So the notation is a ...
3
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1answer
27 views

Show that $X$ can be represented as a union of disjoint equivalence classes

Let $X$ be a set. Let $\sim$ be an equivalence relation on $X$. Show that $X$ is the union of disjoint equivalence classes $\{x\}$ for $x \in X$. What I have tried: claim: $X = \bigcup_{x\in X} \{ x ...
1
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0answers
41 views

Ideas for approaching set theory when you've already studied higher abstractions?

I've come to acknowledge (or so I think). That many of the concepts e.g. in real analysis (like, say, continuity), actually don't (in modern times) boil down just real analysis. But rather, e.g. set ...
0
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1answer
29 views

Set theory; sets and subsets; Is an empty set contained within a set that contains real numbers? [duplicate]

Here is the context for my question: Let A = {1,2,5,8,11}. Here is my question: Is ∅ ⊆ A? Why or why not?
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1answer
35 views

Evaluate the image of a function

I am given a function: $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}, \space\space f(x,y)=(x-y^2)(y-x^2)$. I have to evaluate the image $f(A)$ of a set $A=\mathbb{R}^+\times \mathbb{R}^+$. I tried ...
0
votes
1answer
34 views

What does it mean for the empty set to be connected and totally disconnected?

I am trying to prove that the empty set is disconnected, but every single post I can find on this topic is about showing empty set is connected. Recall definition of connected. A set $S$ is connected ...
2
votes
7answers
54 views

Show that $A \subset B \implies A \cap B = A$ [duplicate]

I am trying to show that: $$A \subset B \implies A \cap B = A$$ So far I got: $$A \subset B$$ $$A \cap B \subset A$$ $$A \cap B \subset B$$ $$A \cap B \subset A \subset B$$
2
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4answers
44 views

How to show that $A \cup B = B \implies A \subset B$

I am trying to show that $$A \cup B = B \implies A \subset B$$ but I get stuck on: $$x \in A \cup B = B$$ $$x \in B$$
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2answers
23 views

Prove that a denumerable set can be partitioned into two denumerable subsets

I was wondering if this "proof" is sufficient in demonstrating a that a denumerable set $A$ can be partitioned into two denumerable subsets $A_1$ and $A_2$. Let $A$ be a denumerable set and define $A ...
0
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2answers
22 views

Proving distributivity of Complement over union (set theory)

I am trying to prove the following identity: $$M \setminus (N \cup L) = (M \setminus N) \cap (M \setminus L)$$ I thought about saying that $x \in (N \cup L)$ which means that $x$ is in either $N$ or ...
0
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2answers
24 views

If a set $S$ has a proper subset $A$ that is infinite, then $S$ is infinite.

I am working through Herstein's Algebra and I have become stuck on a seemingly simple exercise. I am using the definition that a set S is said to be infinite if there exists a bijection between $S$ ...
1
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1answer
24 views

Calculate the number of equivalence relations $S$ that satisfies $R \subseteq S$

Let $A=\{1,2,3,4,5,6,7,8\}$ and let $R=\{(1,2),(5,4),(4,5),(6,2),(4,4),(6,5),(7,8)\}$ be a relation on A. What it the number of equivalence relations $S$ that satisfies $R \subseteq S$ I know what ...
1
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3answers
114 views

What do these symbols mean: $\bigcap$, $\bigcup$, $\bigwedge$, $\bigvee$?

I know that some of these symbols are used in set theory like $A \cup B$, but that's not what I'm talking about. I have seen those symbols used in a way similar to $\Sigma$ summation and $\Pi$ ...
1
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1answer
35 views

To calculate by the index of the element its number

How can I specify a formula to calculate by the index of the element its number? Thank you!
1
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3answers
60 views

Does $z\mathscr Ex ∧x\mathscr E y ⇒ z\mathscr Ey$ imply $z\mathscr Ex ⇒ z\mathscr Ey$ if $z$ is arbitrary?

The proof of Theorem 3 regards $z\mathscr Ex$ $\land$ $x\mathscr E y$ $\Rightarrow$ $z\mathscr Ey$ implies $z\mathscr Ex \Rightarrow z\mathscr E y$. But I think it's not correct because the hypothesis ...
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0answers
20 views

showing a 2 variable function is one to one and onto

The function is $H\left ( s,t \right )=\left \langle 2s \cos\left ( 2t \right ),2s \sin\left ( 2t \right ),12s^{2} \right \rangle\text{ with }\left [ 0,1 \right ]\times \left [ 0,2\pi \right ]$ I ...
2
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4answers
48 views

prove there is a unique set $X$ such that every set $Y$, $Y∪X = Y$

For this proof. It seems obvious that $X=∅$ such that for every set $Y$, $Y∪X = Y$ since the $Y∪X$ is just Y. How should I go about this? Let there be sets $X,Z$ Since $Y∪X = Y$ then, {$x| x ∈ Y ∨ ...
0
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1answer
43 views

What result is $\left | \bigcup_{i \in I}A_i \right | =\sum_{i \in I} |A_i|$?

I'm reading a text that uses the following equality for disjoint sets $(A_i)_{i \in I}$: $$\left | \bigcup_{i \in I}A_i \right |=\sum_{i \in I} |A_i|$$ This has to do with disjoint unions, but I'd ...
1
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1answer
87 views

Proof of the Axiom of Choice

This exercise is from Bloch's book and can be found here. Bloch introduces equivalent variations of the axiom of choice where the one that will be proven is stated in terms of functions: AC1 Let ...
0
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1answer
21 views

Can an undefined value be said to be not an element of a defined set?

With our normal axioms of set theory is it proper to say that an undefined value is not an element of a set containing all defined values? As an example we could be asking "Is the final digit in the ...
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0answers
20 views

Boundedness and extrema of set relating to Basel Problem

Essentially looking for a proof check. Although I think my argument might be fine, want some validation because although the statement is rather intuitive I am not sure I am totally convinced with my ...
0
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2answers
32 views

How to describe $\lbrace \mathbf{x}\in \mathbb{R^n}: |x_j|\le1 $ for$ 1\le j\le n \rbrace $ in terms of $x_j=x_j^+-x_j^-$

How to describe the set $A$=$\lbrace \mathbf{x}\in \mathbb{R^n}: |x_j|\le1 $ for$ 1\le j\le n \rbrace $ in terms of $x_j=x_j^+-x_j^-$ where $x_j^+\ge0$ and $x_j^-\ge0$ The answer says: $B$=$\lbrace ...
1
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1answer
44 views

About the method of proving a set is uncountable

The question is: Prove or disprove: The real numbers with decimal representation consisting of all $1$'s is countable. I received the answer like this $$ \begin{matrix} ...
0
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1answer
20 views

Is $[a]_R$ the same as [a]?

Source: Discrete Mathematics with Applications by Susanna Epp Is $[a]_R$ the same as [a]=a/R? Then, $x \in ([a]_R$=a/R) is the same as $x \in ([a]=a/R)$, right?
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0answers
46 views

Prove there is no one-to-one correspondence function $\colon X→ P(X)$

I'm stuck on the following question and would appreciate it if someone could show me how I can prove it. Let $X$ be any set and let $P(X)$ be the power set containing all subsets of $X$. Prove ...
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0answers
31 views

Given $A \subset U$ and $B \subset U$. Prove or disprove $A - B \ = \emptyset \implies A \subset B$ [duplicate]

Given $A \subset U$ and $B \subset U$. Prove or disprove $A - B \ = \emptyset \implies A \subset B$ I think this is the way to do it: $$ A = B \implies A \subset B \implies A' = B' \implies A-B = ...
0
votes
1answer
28 views

Can a finite set that is odd and a finite set that is even have the same number of subsets?

A more clear way of asking it I suppose would be. Supposing a finite set 'S' that is not empty, how would I be go about proving that the number of subsets of S if the total number of elements is odd, ...
2
votes
1answer
47 views

Cardinality of the set of functions $f: A \to B$ where where $|A|=\aleph_0$ and $|B|=2^{\aleph_0}$

Let $X$ be the set of all functions $f: A \to B$ where $|A|=\aleph_0$ and $|B|=2^{\aleph_0}$. Using some cardinal arithmetic, one can show that $|X|=2^{\aleph_0}$. However, I wanted to construct a ...
1
vote
1answer
39 views

Applying union and power set in different orders.

Show that $E$ is always equal to $ \bigcup\{x:x\in \mathcal{P}(E)\} $ but that the result of applying $\mathcal{P}$ and $\bigcup$ to $E$ in the other order is a set that includes $E$ as a subset, ...
1
vote
1answer
55 views

Should it be allowed to apply classical logic to set theory?

It is well known, that the generalized continuum hypothesis isn't provable from the standard axiom system ZFC. GCH (generalized continuum hypothesis). For every infinite set A, there isn't a set M ...
0
votes
1answer
67 views

Isn't x/E = y/ E ⇔ x E y deduced, not just x/E = y/ E ⇒ x E y from (a) x/ E≠ Ø, (b) x/E ∩ y/ E ≠ Ø ⇔xEy?

"Theorem 3. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. Then (a) Each $x/\mathscr E$ is a nonempty subset of $X$. (b) $x/\mathscr E \cap y/\mathscr E \neq \emptyset$ if ...
-1
votes
1answer
28 views

Questions on symmetric difference of events

From a comment on my math overflow question: No, $P(A\bigtriangleup B)=0$ means $A$ and $B$ are essentially the same except in situations that almost surely do not happen. $P(A)=P(B)$ says much ...