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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0
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2answers
35 views

For all sets A, B, and C, does the equality hold

For all sets A, B, and C, does the following equality hold? $A-(B-C) = (A-B) - C$ $A\cap (\bar B\cup C) = (\bar B \cap A) \cap C$ by DeMorgan's From this, I am able to obtain $A=$ on the ...
1
vote
1answer
48 views

Can the cardinality of a strictly ordered set exceed the cardinality of the natural numbers?

I'm putting some thought into the CH at the moment and a proof of the answer to this question would be most helpful if anybody would be so kind as to help me out: Can the cardinality of a strictly ...
-3
votes
0answers
47 views

Help in Verifying a Set Theory Proof [on hold]

let X and Y be algebras of subsets of disjoing sets M and N, respectively. prove that $ X∪Y= \{{A∪B:A∈X, B∈Y}]$ is an algebra of subsets of the set $M∪N$ 1) $(M∪N)∈(X∪Y) $ $∵M∈X, N∈Y$ 2) ...
0
votes
0answers
38 views

Sets relations through the properties of their elements [on hold]

Let's have two sets of objects, namely $X$ and $Y$. Iterating over the elements of $X$ a software should create new elements in $Y$, whose count should be dependable on the value of a particular ...
5
votes
2answers
67 views

Why can't you count up to aleph null?

Recently I learned about the infinite cardinal $\aleph_0$, and stumbled upon a seeming contradiction. Here are my assumptions based on what I learned: $\aleph_0$ is the cardinality of the natural ...
3
votes
2answers
59 views

simple proof for principle of pigeons

I must prove the principle of pigeons but the proofs I find in the internet are too complex. Here's what I can use: Definition $$I_n = \{p\in \mathbb{N}; p\le n\}$$ The principle of the pigeons ...
0
votes
1answer
60 views

Are these statements “truly” equal?

Consider a set $A$, elements $x,y$ in $A$ and the following propositions: \begin{equation} \exists x\in A\ |\quad x=x \end{equation} \begin{equation} \forall x\in A:\quad x=x \end{equation} ...
1
vote
1answer
50 views

finding sup and inf of $\{\frac{n+1}{n}, n\in \mathbb{N}\}$

Please just don't present a proof, see my reasoning below I need to find the sup and inf of this set: $$A = \{\frac{n+1}{n}, n\in \mathbb{N}\} = \{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, ...
0
votes
0answers
27 views

Proving $\sup A = -\inf(-A)$ [duplicate]

In order to prove that $$\sup A = -\inf(-A)$$ obs: $A$ is limited I did: think of $\sup A$ as a number $$a\le \sup A, \forall a\in A \implies -a\ge -\sup A, \forall a\in A$$ Doesn't that implies ...
0
votes
0answers
30 views

Statements dealing with variables independent of bound variables in a quantifier

I am working on a proof now, but it hinges on whether or not the following equivalence holds: $$\forall i\epsilon I (P(x)) = P(x)$$ Where i and x are independent of each other, x is not necessarily ...
0
votes
0answers
18 views

Existence of convergent sequence in a closed set?

Is it true that if $C$ is a closed set, and $x \in {C}$ then there exist a sequence $\{x_i\}$ which tends to $x$? I'm not sure about the correctness of this statement, and whether it is true for ...
1
vote
5answers
51 views

Help with set proof: $A \cap B = A $ if and only if $A \subseteq B $.

$A \cap B = A $ if and only if $A \subseteq B $ It's been a while since I've done this sort of proof. I can't think of how I would prove this statement. I'm too used to numerical proofs. What ...
0
votes
1answer
21 views

Set builder notation: defining the number of elements

I have a set L and I have a subset S which is part of L and contains three elements A, B and C. Finally, each of these elements are subsets that consist of their own elements: $A=\{a_1...a_n\}$ ...
3
votes
1answer
48 views

If $p_n$ is the $n^{th}$ prime, is it ever appropriate to speak of $p_{\aleph_0}$?

If $p_n$ is the $n^{th}$ prime, is it ever appropriate to speak of $p_{\aleph_0}$? I'm no math student. Your pardon if this is just some clearly obvious and easy answer, I'm just not seeing it. ...
0
votes
0answers
26 views

How can I correctly catalog this partition problem?

Studying the partition problems, I tried to do an special version to apply it to a kind of model of "orbits and energy levels" (explained below), but I am having problems to properly catalog this. ...
4
votes
5answers
66 views

Prove that $A \cup B = A \cap B$ if and only if $A=B$. [duplicate]

Prove that $A \cup B = A \cap B$ if and only if $A=B$. My method was: We must prove two implications, so we will proceed by proving the first implication. We will do this by proving the ...
0
votes
1answer
63 views

Mapping an element from a set X to a pair of elements from a set Y through a property

A software iterates over existing set of objects $X$ and for each element $x \in X$ creates two elements (non-existing by now objects) in the co-domain $Y$. The index relation is such, as element ...
0
votes
0answers
23 views

How to show that an $\omega$-sequence is club in a limit ordinal $\gamma$.

Suppose that $\gamma$ is a limit ordinal. We say that a subset $X \subseteq \gamma$ is closed if whenever $\delta < \gamma$ is a limit ordinal and $\sup(X \cap \delta) = \delta$, then $\delta \in ...
1
vote
0answers
30 views

Show a set is countable

Let the set T(8) = {all subsets containing 8 elements of $\mathbb{N}$}. Show that T(8) is countable. Proof: $T(8)=\mathop{\cup}\limits_{n=1}^\infty\{\{n,x_{2},...,x_{8}\}:x_{2},...,x_{8}>n\} = ...
4
votes
0answers
59 views

Deducing the existence of particular functions $\mathbb{N}\longrightarrow\mathbb{Q}$ in the context of Tom Leinster's “Rethinking Set Theory”

This question concerns the set theory given by Tom Leinster in his paper "Rethinking Set Theory," available here: http://arxiv.org/abs/1212.6543 In this paper, axioms for set theory are given in the ...
-3
votes
0answers
25 views

is there exist a partial order set with no maximal chain? [closed]

let A be a partial order set can this set have no maximal chain ? and if it does give me an example
-4
votes
0answers
11 views

does there exist a choice function always for a family of set? [closed]

there exist a choice function always for a family of set is that true or false and if it is true what is the prove and if it is false give me an example
0
votes
2answers
57 views

How can I prove that these sets are countable infinite?

I am very new to proofs so please excuse any trivial errors. In lecture we were told that: A set $\mathbb S$ is called finite if there exists a one-to-one mapping (bijective mapping) between ...
-4
votes
1answer
34 views

Some things in set theory [closed]

i want some links to these things in set theory that have good explanation and examples if can, i tried searching and didnt find TY all Cantor's theorems in set theory Schröder–Bernstein theorem ...
6
votes
3answers
1k views

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 ...
1
vote
2answers
24 views

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 ...
0
votes
0answers
27 views

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 ...
-1
votes
0answers
29 views

How can we show that $R:=\{ x:x\notin x \}$ is a proper class? [closed]

Is this sufficient? If $A=\{ x:x\neq \emptyset \}=V-\{ \emptyset \}$ By the union axiom, $V$ is a set and so too is its powerset. $P(V)>V$ by Cantor's theorem.
1
vote
3answers
27 views

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 $f$ ...
2
votes
2answers
19 views

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. ...
-1
votes
1answer
22 views

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.
0
votes
1answer
23 views

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 ...
0
votes
0answers
28 views

Sets vs Objects, is there a difference [closed]

I'm just curious, is there a difference between the mathematicians set and the programmers class? It's seems like a class of N categories is just an n-tuple set?
0
votes
2answers
26 views

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 ...
-2
votes
1answer
31 views

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?
0
votes
1answer
37 views

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 ...
1
vote
2answers
33 views

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 ...
1
vote
0answers
33 views

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 ...
2
votes
0answers
54 views

Carother's “certainly” proof about measurable sets

Carother's Real Analysis text has the following Theorem. Can someone check if my proof is correct? $(i \Rightarrow ii)$ Let $E$ be a measurable. Let $I_k$ be open intervals, such that $$m^*(E) ...
0
votes
2answers
23 views

Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
1
vote
1answer
72 views

Should $\bigcap_{n = 1}^\infty (a-\frac{1}{n}, b + \frac{1}{n})$ be $(a,b)$ or $[a,b]$

I am confused about the limiting behavior of as $n \to \infty$, $\bigcap_{n = 1}^\infty (a-\frac{1}{n}, b + \frac{1}{n})$. I have read that it is the case that this set becomes closed, but I can't ...
0
votes
1answer
30 views

Could multiplication be defined as $|A_1 \cup A_2 \cup ..A_n|$?

Taking discrete math which suddenly right now I am thinking of what dividing and multiplication do when it comes to what I learned in this class. But would multiplication be defined as the cardinality ...
0
votes
0answers
28 views

mathematical formula to compute sum of all sub sequences of a number N

We have a number say N and we list down all its sub- sequences and sum them up.SAY for n=123 ,the sum is 177(123+12+23+13+1+2+3). I came across this mathematical formula which computes the sum taking ...
-1
votes
1answer
27 views

Prove set relations without using Venn Diagrams

For sets A and B,(A\B) = A∩B^c show that (A\B)\C ⊆ A(B\C) without using Venn Diagrams show that A∩B and B\A are disjoint without using Venn Diagrams Need explanation to solve this sum. Cheeers!
1
vote
2answers
43 views

Proof of a property of set of all one-to-one mappings

Let $S$ be a nonempty set and $A(S)$ be the set of all one-to-one mappings of $S$ onto itself. I.N. Herstein in Topics in Algebra says (in page 28) that whenever $S$ has three or more elements, we can ...
1
vote
1answer
29 views

How to verify this relationship between area under the graph and the preimage?

Define $h : \mathbb{R} \to [0, \infty)$, Let $H = \{(x,y)| 0 \leq y \leq h(x)\}$ be the area under the graph (including the boundary) I wish to show the following is true: $$H = ...
1
vote
2answers
23 views

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 ...
0
votes
0answers
29 views

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 ...
0
votes
1answer
28 views

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 $\mathcal{P}(S)$. ...
0
votes
3answers
83 views

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 $ ?