# 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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### What does multiply mean in a set

I have seen this question today. $[A \cdot(B-C)]\cap[(A \cup C)\cdot(C-B)]$ 1.$(A-B) \cdot (A-C)$ 2.$(A \cdot B)\Delta (A\cdot C)$ 3.$(A \cup B)\cdot (A \cup C)$ 4.$\emptyset$ ...
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### Is “closedness” a proper word?

In one of my papers I had to prove a list of properties of a set, say, $S=\{a,b,c\}$. Among them we have a fact that $S$ is downward closed with respect to a binary relation $R$. I found it awkward to ...
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### Transitive Property of Proper Inclusion

$Theorem:$ If $A \subset B$ and $B \subset C$, then $A \subset C$. Here $X \subset Y$ is defined as $X\subseteq Y$ and $X\neq Y$. I can prove the case of improper inclusion using the ...
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### About Cantor's proof of uncountability or real numbers

The proof is using reductio ad absurdum , i.e. contradiction. Start with that there is a sequence of all real numbers (in some interval) and then it is shown that there exists number that is different ...
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### Function over non-numerical sets

Considering a finite lexicographically ordered set, for example, $\{a, b, c, d\}$ called $A$ with $A$ as domain and codomain of a function which returns the element with right shift of 1 over A, how ...
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### Explaining the proof that the set of integer combinations of a,b is equal to the set of integer multiples of the GCD of a,b

I'm trying to understand how this proof works, but the website is using a different notation than my book and it's not making any sense to me. here is the proof on ProofWiki I know that the ...
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### My proof that if $P(A) \subseteq P(B)$, then $A \subseteq B$

I'm not sure if my proof is sound. Here it is: Assume that $P(A) \subseteq P(B)$, so any subset C of A is also a subset of B. Therefore, any element in C is also an element of A, and by the same ...
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### Unable to prove that a statement is an equivalent form of the Axiom of Choice [duplicate]

The exercise 22 page 158 of Elements of Set Theory by B. Enderton is the following: Show that the following statement is another equivalent version of the axiom of choice: For any set $A$ there ...
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### Checking reflexive, symmetric and transitive properties of $\neq$ on $\mathbb{N}$

QS: Indicate if the relation on the given set are reflexive on a given set, which are symmetric, and which are transitive. $\not = \text{on } \Bbb N$ So for this problem I am trying to ...
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### prove sets cardinality inequality

I need to prove that if $$A , B$$ are infinite sets and it holds that : $$|A| > |B|$$ then: $$|A \backslash B| = |A|$$ I guess I just don't what can I say about the cardinality of |A\B| ...
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### What branch/field of mathematics is this? [closed]

I do not want solutions, I just want the field/branch of mathematics that these problems deal with, and possibly a good online source or two to learn it. Problems :- 1:- 2:- 3:- 4:- ...
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### What is the complement of a product of two sets?

I am given this information: Suppose $A=\{1,2,3\}$, $B=\{3,5\}$, $C=\{1,2,4,6,9\}$ and $U = \{0, 1, 2, 3, 4, 5, 6,7,8,9\}$. Enter "T" for each true, and "F" for each false statements. There ...
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### What are example of minimal but not least element? [duplicate]

I know that every least element are minimal. But, What are example of minimal but not least element?
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### How do you prove that the greatest element of a partially ordered set (A,<=) is maximal? [duplicate]

1) How do you prove that the greatest element of a partially ordered set (A,<=) is maximal? 2) What will be some example that the maximal is not necessarily a greatest element?
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### How do you prove $X\times Y = \varnothing \iff X=\varnothing$ or $Y=\varnothing$? [closed]

How can I prove that these two are equivalent? $X\times Y = \{(x,y)\mid (x\in X)\land(y\in Y)\} = \varnothing$ $X=\varnothing$ or $Y=\varnothing$.
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### Cardinality question on set of symbols [closed]

Few moments ago I asked myself a question, that I not positive if, in fact, is well defined. Let $\mathbb{R}$ be the set of real numbers. Define $S$ to be a set of symbols, as follows: Let $x$ be ...
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### uncountability proof [closed]

In the proof that the set of all sequences whose elements are 0 and 1 if we replace the set of reals with the set of natural numbers wouldn't that lead to the same contradiction that N is a proper ...
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### Calculate maximum value limit in set partition

I have all the subset of 4 element as follows with value associated with each subset ...
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### how to write a set of local extrema formally

Is the following notation correct? $X = \left\{x_t > \max\left(x_{t-z},\ldots,x_{t-1},x_{t+1},\ldots,x_{t+z}\right) \mid (t-z, \ldots, t+z)\in \mathbb{T}^{2z+1} ∧ k ∈ ℕ\setminus \{0\}\right\}$ ...
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### How do I prove αβ = 0 ⇐⇒ α = 0 or β = 0?

Let α and β be cardinal numbers. Prove that αβ = 0 ⇐⇒ α = 0 or β = 0. Below is what I have done. Could you please review and point out illogical parts, and things I have missed out? I want to be ...
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### Prove that $\bigcup\mathbb{N}=\mathbb{N}$

Prove that $\bigcup\mathbb{N}=\mathbb{N}$. Showing that $\mathbb{N}\subseteq \bigcup\mathbb{N}$ is simple. However, I'm not seeing how to handle showing $\bigcup\mathbb{N}\subseteq\mathbb{N}$. I ...
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### Is there a bijection from $A = ]0,1[$ to $B = A \cup \{1,2,3,4\}$?

Is there a bijection from $A = ]0,1[$ to $B = A \cup \{1,2,3,4\}$? If there is, give example. I had this question on exam few days ago, and I have been googling for days, but I can't find a ...
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### Exercise from David Williams' “Probability with Martingales” page 16

I am reading David Williams' "Probability with Martingales" and I'm completely stuck at the exercise page 16: Let $\mathcal{C}$ be the class of subsets $C$ of $\mathbb{N}$ for which \lim_{m \...
I am trying to solve the following problem. We would like to construct $\{A_1, \ldots, A_n\}$, where $n$ is even, and each $A_i \subseteq [m]$, with $|A_i| = k$ and $m = \text{poly}(n)$. Now, I would ...