# 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, ...

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### Real numbers as element of a universe

defn. A universe is a set $U$ such that: $x\in u\in U\Rightarrow x\in U$ $u\in U$ and $v\in U$ imply $\left\{u,v\right\}, \langle u,v\rangle, u\times v\in U$ $x\in U\Rightarrow \mathcal{P}(x)$ and ...
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### Show that $\bigcup_{n=1}^\infty A_n= B_1 \backslash \bigcap_{n=1}^\infty B_n$

Let $\{B_n\}$ be a decreasing set $B_1 \supseteq B_2 \supseteq B_3 \supseteq ....$ Define $A_n = B_1 \backslash B_n$ i.e. $A_1 = \varnothing, A_2 = B_1 \backslash B_2$ If we imagine $\{B_n\}$ as a ...
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### Is it possible to define a set $S=\{ x \ |\ 1<x \le 1\}$?

I was wondering what would happen if I defined a set $S$ like this $S=\{ x\ | \ 1<x \le 1\}$. My main question would be if defining $S$ like that would be valid in set theory or if it is ...
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### How to phrase a proof of a function from a set A to a set B

Here is a problem: Let $f \subseteq A \times B$ be a function. In many situations you may want to restrict the domain of $f$ or expand its range. If $C \subseteq A$ then define the restriction of $f$ ...
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### finding counterexample for identity of sets

Let $A_i,B_i,C_j,D_j$ be sets. I am wondering if the equation $$\bigcup_{i\in I}(A_i\times B_i) \cap \bigcup_{j\in J}(C_j\times D_j)=\bigcup_{i\in I,j\in J}(A_i\cap C_j)\times (B_i\cap D_j)$$ holds. ...
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### set theory simplificaiton (laws of set theory)

I am relatively new to set theory,and I have to simplifiy this $$(Y \setminus X) \cap X$$ I'm stuck, can anyone help me.
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### minmum number of subsets of $\{1, 2, 3, … , n\}$, each of cardinality $r$, required such that their intersection is $\{1, 2, 3, … , m\}$

Let $M = \{1, 2, 3, ... , m\}$ and $N = \{1, 2, 3, ... , n\}$ be sets with $m < n$. Let $r \in \{1, ... , n\}$, with $m < r$. What is the minmum number of subsets of $N$, each of ...
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### Prove by contradiction that If $R$ is a transitive relation on set $A$ then $R^2$ is transitive.

I saw this problem and read through it but I am still kind of confused as to what $u_1$ and $u_2$ stand for. Prove by contradiction that for a transitive relation $R$ on $A$, $R^2$ is also transitive ...
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### A = { x:x²=1, x is integer} How to get -1?

A = { x:x²=1, x is integer} How to get -1? The answer is -1 and 1. 1² = 1 But where -1 come from?
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### How to prove something is an equivalence class?

I don't understand equivalence class and representative function: http://www.cdhmhome.com/uic/math215/S.pdf , I'm looking at examples 37, 1-7 and have already determined which ones are equivalence ...
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### Is there a continuous function $f:X\rightarrow Y$ such that $f\big(\cap_{i\in I} A_i) \neq \cap_{i\in I}f\big(A_i)$

Is there a continuous function $f:X\rightarrow Y$ such that $$f\big(\cap_{i\in I} A_i) \neq \cap_{i\in I}f\big(A_i)$$ Where $A_i\subseteq X$ and $I$ is an arbitrary index set. I can easily find a ...
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### I'm looking for a good book on FOL and set theory.

I finally decided to really learn some axiomatic set theory, at least the basics. I've studied a bit of FOL, but a review would be nice. In short, I'm looking for a book that focuses on $\sf ZFC$ or ...
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### Counting number of relations that are symmetric and reflexive.

I've looked at the other two problems similair to mine but I'm having a bit of an issue understanding as their solutions seems a bit more complex. While I for the most part understand my professors ...
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### Equivalence Relations and Cardinality

I'm looking at the question below from a past paper: What is an equivalence relation? Say that two sets $X$ and $Y$ are related via the relation $\rho$ if $X$ and $Y$ have the same cardinality. Prove ...
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### Clarification needed for a set notation

I have a question with regards to this problem: Let $S = \{1,2,3,4,5,6\}$ and let $P(A): A \cap \{2,4,6\} = \emptyset$ and $Q(A): A \neq \emptyset$ be open sentences over the domain $\mathcal{P}(S)$. ...
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### Given $3$ sets if $A \cap B = A \cap C$. Is $B=C$? [closed]

I've been trying to solve this question for a while but I just seem to be missing something. Let $A$, $B$ and $C$ be three sets. If $A \cap B = A \cap C$ Is $B=C$ ?
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### Let $A$ be an infinite set and let $B$ be a set such that $A$ is equinumerous to a subset of $B$. Then, $B$ is infinite.

To me, the proof is as simple as this: Let $C\subset B$ such that $A\sim C$. Then, as $A$ is infinite, we have that $C$ is infinite. Thus, as $C\subset B$, it must be that $B$ is infinite. Thus, $B$ ...
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### How many antisymmetrical relations are there on a set $B =\{ 1,2,3\}$? [duplicate]

How many antisymmetrical relations are there on Set $B$ if Set $B = \{1,2,3\}$? I believe its three?
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### What is meaning of canonical form of a set?

Let A{1,2,3}, B1={1,2}, B2={2,3}. Find minset canonical form of B1 and B2. We know minsets generated by B1 & B2 are A1 = B1 \cap B2^{c} A2 = B1 \cap B2 A3 = B1^{c} \cap B2 A4 = B1^{c}\cap B2^{c} ...
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### Is this reflexive, symmetric, antisymmetric or transitive?

Set A contains all points $(x, y)$ on a coordinate plane. The relation $R$ is defined as: point $(x_1,y_1)$ is related to point $(x_2,y_2)$, if $y_1=y_2$. Is this set (A) reflexive, symmetric, ...
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### 10-Subset Sum: Given a set of integers K and an integer M, is there a subset of exactly 10 elements of K whose sum equals M?

I understand that the more general Subset Sum problem is NP-complete, but I am under the assumption that this more specific version of the problem can be solved in polynomial time. However, I can't ...
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### Injective: In what circumstances would there be less than one pre-image of an image?

I am trying to get my head around the different types of function in set theory. In the definition of injective where: if (x',y) is in f and (x'',y) is in f, then x'= x'' in other words y has no ...
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### Sets and Universes

A universe is a set $\mathcal{U}$ such that: 1) $x\in\mathcal{U}$ and $y\in x$ implies $y\in\mathcal{U}$ 2) $I\in\mathcal{U}$ and $x_i\in\mathcal{U}$ for every $i\in I$ implies ...
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### Can we use the terms 'class of sets' and 'family of sets' interchangeably?

I read in pg-4, Introduction to Topology and Modern Analysis by Simmons that class refers to a set of sets while family refers to a set of classes. I formulated an example for the same - if points are ...
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### Is it true that if $cl(S) = int(S) \cup \partial (S)$ and $cl(S) = S \cup \partial (S)$ then $int(S) = S$ which means $S$ is open?

Like the title says, I know: $cl(S) = int(S) \cup \partial (S)$ But also; $cl(S) = S \cup \partial (S)$ But then $S = int(S)$ which is only true if $S$ is an open set. Where is this limitation ...
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### Let $X, Y$ be topological spaces and let A ∈ B($X$) (Borel $\sigma$ algebra on $X$), B ∈ B($Y$). How to show that A × B ∈ B($X\times Y$)?

Let X be a topological space. All that I know is Borel $\sigma$ algebra on X is the smallest $\sigma$ algebra generated by $T_X$ i.e. set of all open sets in X. Is there any other characterization of ...
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### Is directed set countable, if for each element there are only finitely many smaller ones?

A directed set is a pair $(A,\leq)$ where $\leq$ is a reflexive, transitive relation such that for any $x,y\in A$ we have some $z$ such that $x,y\leq z$. (This comes up when dealing with categorical ...
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### Standard notation for 'same' function with different ranges

Does anyone know of a standard notation for the situation when we want to define the 'same' function but on a larger or smaller range. More precisely, if $$f:A \to B$$ is a function and $C$ contains ...
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### Comparability of a set and a subset of power set.

It's well known that for any set $A$, $A < P(A)$. But now, I have some question that, WITHOUT AC, can we guarantee that $A \leq X$ or $X \leq A$ whenever $X \subseteq P(A)$? Thank you.
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### The number $\binom{8}{4}$ is equal to the number of subsets of size 4 of the set $\{1, \dots, 8\}$

I was asked to proof if is true and give a counter example if it is false. However I prefer True. since all the numbers 1-8 insides the brackets are in the sets. I'm I correct?
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### Finding equivalence classes in a set of functions with a condition

I can't seem to work around this problem... Let $n$ and $m$ be two positive integers. $F$ is the set of functions from {1,..,n} to {1,...,m}. We define the relation $R$ as: $f R g$ if and only if ...
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### De Morgan's Laws proof

My only problem with this proof is that they seem to be assuming that $(A \cup B)'$ is nonempty (similar idea for $A' \cap B')$. Is that because this holds trivially if $(A \cup B)'$ is empty? In ...
Given a set $\{1,2,...,n\}$, how would one go about finding the number of subsets that contain the number 2, for example. Having had a play around with this myself I've got as far as realising that ...