# 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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### Question about identities and bijective functions

Let $S$ and $T$ be sets and let $f : S \to T$ be a function. Prove the following: If $g,h:T \rightarrow S$ are functions satisfying $g \circ f =Id_S$ and $f \circ h=Id_T$ then $f$ is bijective ...
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Let $S$ and $T$ be sets and let $f : S \to T$ be a function. Prove the following: If $R$ is a set and $h:R \rightarrow S$ is a function such that $f \circ h$ is surjetive then also $f$ is ...
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### If A is finite, then P(A) is finite.

I am solving the following exercise from Munkres. I was able to prove part (a) by the use of indicator function I proved that $\phi : P(A) \rightarrow X^n$, where $\phi(B) = I_B$. Here $I_B$ is the ...
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### $A \Delta C = B \Delta C$, then prove that $A = B$ where $\Delta$ is a symmetric difference operation.

I suppose that if I can prove that every element that belongs to set $A$ also belongs to set $B$ and vice versa and also any element that doesn't belong to set $A$ doesn't belong to set $B$ either and ...
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### (1) let ans = empty (2) for i from 1 to n do: $ans = A \cap ( ans \cup B_i )$ How to prove that $A \cap (\cup B_i) = ans$?

I have a problem on set theory. My problem is: (1) let $ans_0 = empty$ (2) for i from 1 to n do: $ans_i = A \cap ( ans_{i-1} \cup B_i )$ How to prove that $A \cap (\cup B_i) = ans_n$ ?
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### what does “a set of sets that are not members of themselves” of Russell’s Paradox mean

Russell’s Paradox begins with a statement of "Let $R$ be the set of sets that are not members of themselves", i.e. $R=\{S\mid S\notin S\}$. I'm a little bit confused with the statement, for example, ...
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### $S$ be $\pi$-system on a set, given two measures on $\sigma(S)$, is there a topology on $\sigma(S)$ making $S$ dense, and the two measures continuous?

Let $\Omega$ be a non-empty set , $S \subseteq \mathcal P(\Omega)$ be a Pi system (https://en.wikipedia.org/wiki/Pi_system ) on $\Omega$ , let $\sigma(S)$ be the $\sigma$-algebra generated by $S$ (i.e....
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### How do I find the powerset of $A\cap B$?

$A = \{0,1\}$ $B = \{1,2\}$ My Working : $P(A\cap B)= P(\{\varnothing, \{1\}\}) = \{\varnothing,\{1\},\{\varnothing,\{1\}\}\}$ But the correct answer is $P(A\cap B)= \{\varnothing , \{1\}\}$.
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### Finding an example of a bijection from $\Bbb N$ to $E^+$.

Give an example of a bijection $h$ from $\Bbb N$ to $E^+$ such that $h(1) = 16, h(2) =12, \text{ and } h(3) = 2.$ $\Bbb N = \text{ natural numbers }$ , $E^+= \text{ positive even integers. }$ So ...
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### Without AC is there a relationship between $\beth$ and $\aleph$ numbers?

Assuming AC we know that all $\beth_\alpha$'s will be $\aleph_\beta$ for some $\beta$ since they can be well ordered. Can anything interesting be said about their relationship without AC? Is it ...
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### Is the set of aleph numbers countable?

If I write the set of aleph numbers in this way $\{\aleph_0, \aleph_1, \aleph_2, \aleph_3, \dots\}$ it seems obvious to me that this set is countable, because aleph numbers have integer coefficients. ...
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### P(P(N)) is equinumerous to the set of real functions

I need to show that (the power set of power set of N) P(P(N)) is equinumerous to the set of real functions. I know that P(N) is equinumerous to R, thus it is equivalent to show that $\{0,1\}^R$ is ...
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### Let $R$ be a relation on a set $A$. Show that if $R \circ R \subseteq R$, then $R$ is transitive

On a recent quiz I encountered the problem: Let $R$ be a relation on a set $A$. Show that if $R \circ R \subseteq R$, then $R$ is transitive. I gave the following answer: Assuming $R$ is a relation ...
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### Question on finite subfamilies of an infinite family of sets

Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$. Suppose $\bigcap X\subseteq B$. Is it ...
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### Proving transitivity of a relation

Let R be a reflexive relation on a nonempty set X. The asymmetric part of R is defined as the relation $P_r$ on X as $xP_ry$ iff $xRy$ but not $yRx$. The relation $I_r$ = $R\setminus P_r$ on X is ...
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### How Can You Define Successors Over The reals?

I've been going through Introduction To Modern Set Theory by Judith Roitman, and am confused by her exposition of well orderings. She gives the following proof that every element $x$ of a well-ordered ...
### Topology: What does sets in $\bigcup_{i \in \mathbb{N}} A_i$ look like?
Let $\tau$ be the topology on some set $Y$, and $f_i: X \to Y$ be some continuous function. Let $A_i = \{f^{-1}_i(U)| U \in \tau\}$ and $A = \bigcup\limits_{i \in \mathbb{N}} A_i$ My question is ...
How do I prove that for any finite subsets A and B exists one set R $\left | A\cup B \right |=\left | A \right |+\left | B \right | -\left | A\cap B \right |$ Deduce from this an adequate formula ...