This tag is for elementary questions on set theory - the sort of material covered in "Chapter 0" of graduate texts and in undergraduate set theory texts. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, countability and uncountability, power sets, ...

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0
votes
2answers
26 views

How to solve this question subset

Answer true or false to each of the following questions. If a statement is true, prove it. If a statement is false, give a counterexample. For all sets $A$,$B$ and $C$: IF $A ⊆ B$ and $A ⊆ C$, Then ...
4
votes
3answers
386 views

Definition of set.

A set is defined as a collection of distinct objects. Why have we defined a set to contain only distinct objects? Why is a collection of objects which may have identical objects not called a set? ...
3
votes
3answers
245 views

Is every set a pointed set?

My question is quite simple, It seems every non-empty set is a pointed set, only we have to do is choice some element to be the distinguished element, am I right? I'm looking for non-empty sets which ...
0
votes
2answers
17 views

Proving that if $E, F$ are equivalence relation on $A$ and $E \subseteq F$, then there is a surjection from $A\setminus E$ to $A\setminus F$

Proving that if $E, F$ are equivalence relation on $A$ and $E \subseteq F$, then there is a surjective function from $A\setminus E$ onto $A\setminus F$. What does $E \subseteq F$ even mean? Does it ...
-7
votes
1answer
60 views

Countablity of the set of the points where the characteristic function of the Cantor set is not continous

We are creating the Cantor set typically starting from the interval $[0,1]$ and removing $\frac{1}{3}$ of it like it is described here or here. The problem is to resolve if the set of discontinuities ...
-2
votes
2answers
36 views

Set theory (A\C) \ (B\C) = (A\B)\C [duplicate]

Very difficult question. Can't make sides equal after multiple manipulations.This one is lost on me. Help a fellow human out. Thank you.
2
votes
1answer
31 views

What axiom makes it possible to take the union or intersection of an infinite number of sets $A_1, A_2, \ldots,$ and get a resulting set $B$.

What axiom makes it possible to take the union or intersection of an infinite number of sets $A_1, A_2, \ldots,$ and get a resulting set $B$. In probability I've calculated $P(\cup_{i=1}^{\infty} ...
1
vote
1answer
44 views

What's the negation of “E is uncountable” ??

I think "E is countable or finite." But when I asked my professor, he said "E is countable."
-1
votes
1answer
53 views

Hausdorff topologies on the natural number set are sigma algebra

Is it true that if I add the Hausdorffness condition to any topology on $\mathbb{N}$, then it is a $\sigma$- algebra on $\mathbb{N}$? Once I have tried to prove this, I think that compactness is also ...
2
votes
3answers
35 views

Empty intersection of empty sets [duplicate]

In what sensible way can we define the union and intersection of an empty family of sets? How come empty intersection of empty sets become the whole space ?
1
vote
3answers
53 views

Showing a subset is uncountable [closed]

How do I show if $A \subseteq B$, and $A$ is uncountable then $B$ is uncountable?
0
votes
1answer
27 views

Cardinality of the union of all repeated Cartesian products of N with itself [duplicate]

Here is a silly question, but I am a silly person. Consider the: Natural Numbers. Natural Numbers X Natural Numbers. Natural Numbers X Natural Numbers X Natural Numbers ... Now take the union of all ...
2
votes
5answers
125 views

Help me understand 'equivalence classes' and relations

I'm reading up on binary relations and I understand them to be a mapping from one set into another. However I'm having problems understand 'equivalence classes'. My book only gives a pretty dry ...
7
votes
2answers
189 views

cartesian product $A^2 = A$, possible?

Do there exist non-empty sets $A$ such that $A\times A = A$? $A\times A = A$ looks a little strange to me, since $A\times A$ seems somehow more complicated than $A$, hence it is unlike that they are ...
2
votes
4answers
62 views

Why does $B^A$, not $B\cdot A$, define set of all functions from set $A$ to set $B$?

Let $A$ and $B$ be sets. Define $A = \{a, b\}$ and $B = \{x, y, z\}$. According to my readings, the set of all functions from $A$ to $B$ can be defined by $|B|^{|A|}$. This would dictate that there ...
1
vote
1answer
28 views

Problems with the usage of Belief and Common Belief operators

I have a problem with the usage of a Belief operator $B_i$ in the derivation of a result on a common belief operator $CB$. First of all, some basic definitions (where $i$ is an individual), that ...
1
vote
1answer
28 views

Proving a Bound for Oddtown-Eventown or Clubtown

Suppose we have a town with a set of residents $V$, where $|V| = n$. The residents like forming clubs, and we have clubs $C_1,C_2,\ldots,C_m \subseteq V$. We are interested in the maximum number of ...
0
votes
2answers
30 views

Writing in set builder notation

I need to write following set in to set builder notation. $\{ 0,3,6,9,12 \}$ Solution which I found on internet is: $\{3x\;|\;\text{where }x\text{ is an integer and }0\leq x \leq 4\}$ What I ...
-1
votes
1answer
43 views

Did I prove this correctly? [closed]

I used De Morgan's law. Is that correct...? If not, please give me some hint T.T
1
vote
1answer
16 views

Set Question - is this possible?

Is it possible to have a set $Y$, such that: $Y \subset \mathbb{R}^n$ $\forall y \in Y, \forall\alpha > 1, \alpha y \in Y$ $Y$ is non-empty and bounded. Is such a set an impossibility? Thanks. ...
1
vote
2answers
26 views

Is it possible for a non-well ordered but linearly ordered set to have a proper $<$-inductive subset?

Say we have a set $S$, which is linearly ordered, but not well ordered. Let $B$ be a subset of $A$ such that $\forall t\in A$, $$\text{seg }t\subseteq B\implies t\in B$$ Here, $\text{seg }t=\{x\in ...
1
vote
2answers
25 views

Prove that only one of $x<y,x=y,x>y$ can hold.

I have to prove that in a partially ordered set, only one of $$x<y,x=y,x>y$$ can hold. My book says if both $x<y$ and $x=y$ hold, then this will imply $x<x$, which is a ...
0
votes
1answer
31 views

Union of preimages [duplicate]

Given $f:X\rightarrow Y$ as a function, the image of $x$ if $f(x)$. The preimage of $y$ is $f^{-1}(y)=\{x\ |\ f(x)=y\}$, with the symbol PreIm$(Y)$ Given the definition, could you prove the following ...
3
votes
2answers
37 views

When to use $\in$ or $\subseteq$?

If I have a family of $n$ sets like this $\mathcal{F}=\{\{S_1\}, \{S_2\}, \dotsc, \{S_n\}\}$. What is the right notation: for some $i\;\{S_i\}\in\mathcal{F}$ or $\{S_i\}\subseteq\mathcal{F}$? In ...
0
votes
1answer
45 views

Cardinality of the set of all functions from A to B

Given two sets $A$ and $B$, let $F(A, B)$ denote the set of all functions $f : A → B $ (no assumptions about injectivity or surjectivity – all functions from $A$ to $B$ are included). Let $|S|$ denote ...
2
votes
2answers
68 views

Is this set uncountable or countable?

I have to prove whether or not this set is countable: the functions from $\mathbb Z$ to $\mathbb R$ such that $f(n)=0$ except for a finite number of $n \in \mathbb Z$. I think this set is uncountable. ...
0
votes
0answers
18 views

proof of finite set property

I have a trouble with a problem of proving that "X is finite implies that there exists a function f that maps X to natural number such that f is injective" Can someone help me?
0
votes
2answers
24 views

Find an injective but not surjective function to a infinite set X

A set if infinite iff there exists an injective function f that maps X to X for which f(x) is a proper subset of X. I only know that set X is infinite and I do not know anything about the range of the ...
0
votes
4answers
47 views

Can a transcendental number be an infimum of a set of rationals?

Say I have a set of rational numbers $S \in \mathbb{Q}$. I know that $S$ can have an irrational infimum, for example the set $\{s \in S \:|\: s^3 > 2\}$ has irrational infimum $2^{1/3}$. However, ...
-1
votes
1answer
24 views

Can we construct a chain of all elements greater than $a\in S$ in partially ordered set $S$?

Say we have a partially ordered set $S$, and an element $a\in S$. Can we construct a (possibly infinite) chain of all elements greater than $a$ in $S$? For example, say we have $\Bbb{N}$. If we take ...
1
vote
3answers
46 views

Understanding Set Theory and Proving $A \cap(B\cup A) = A$

I am trying to wrap my head around discrete mathematics in order to help my understanding of self taught programming. I am now trying to understand Set Theory, more specifically proving certain ...
3
votes
3answers
37 views

If two sets are bijective to a common third set, is their union also bijective to that set?

If $S_{1}\xrightarrow{\rm BIJ }\mathbb{N}$ and $S_{2}\xrightarrow{\rm BIJ }\mathbb{N}$, can I conclude that ($S_{1}\cup S_{2})\xrightarrow{\rm BIJ }\mathbb{N}$? It makes sense to me but I'm not sure ...
5
votes
2answers
47 views

Can we only have a Cartesian product of a countable number of sets?

Can we only have a Cartesian product of a countable number of sets? I suspect the answer to this is yes. This is because the resulting tuple of the Cartesian product will be ordered, and each ...
0
votes
1answer
21 views

problem about equinumerosity

How we can prove that $R$ is equinumerous with $R^n , n>1$ ? Of course we can define an injection from the set of real numbers to $R^n$ , but the inverse is the main problem i think in order to ...
4
votes
1answer
30 views

If $2^{\kappa}<\lambda$, how many subsets of size $\kappa$ are there of a set of size $\lambda$.

Assume both cardinals are infinite. Also assume AC as needed. So, the obvious bound is that there are no more than $\lambda^\kappa\leq 2^\lambda$ of them. But it seems there should be an easy bound ...
0
votes
0answers
28 views

Proving a set is equal to another set

For all sets $A$ and $B,(B-A)=B\cap A^C$. I would like to know if this proof is correct or if I am on the right track. Here it is: Let $b \in B$ such that $b \notin A$ than $b \in B$ and $b \in ...
1
vote
2answers
32 views

How to Prove this Set Question? [closed]

For both sets C and D, provide a proof that C ∪ (D − /C) = C "/C" is a set's complement of C
-1
votes
2answers
103 views

Give a Bijection that is in-between 2 intervals and use a formal proof to show that it is a bijection. [duplicate]

∀w,x,y,z ∈ R, w < x and y < z. Given that information, supply a bijection between the two intervals. (w,x) and (y,z) Then after you find the bijection, provide a formal proof that what you found ...
-2
votes
1answer
40 views

For any set A, define the power set P(A) be the set of all subsets of A. [closed]

Prove that there is no bijection from A onto P(A) for any set A.
0
votes
1answer
35 views

How can I prove this power set relation? [duplicate]

Please show me how can I prove this relation: $$P(A \cap B) = P(A) \cap P(B)$$ I have no idea how to prove these kind of relations that include power sets.
1
vote
1answer
23 views

Does for a set of cardinals a finite subset exist such that for any cardinal in the set a larger cardinal in the subset exists?

I am writing an essay for which I need to prove that sufficiently many graphs of a certain type exist. Is it true that for any set of sets (or set of cardinals) $S$ a countable subset $C$ exists such ...
2
votes
0answers
27 views

Maximum and minimum values of intersection of sets

I know how to do this problem, but my question is more on the proving the inequality and the extreme values. So here is the problem: Of the 24 students in a class, 18 like to play basketball and 12 ...
4
votes
1answer
48 views

Confusing verse in “Axiomatic Set Theory” by Patrick Suppes

While searching for prime ordinals, I found this: Goldbach’s Hypothesis is that every even natural number > 2 is the sum of two prime numbers. On the basis of the obvious definition of prime ...
1
vote
1answer
62 views

Set-theoretic notion of differentiation and integration?

Since set-theory is said to be one of the foundations of mathematics, I would think that every mathematical concept is definable in set-theoretic terms, right? How would you define differentiation ...
2
votes
2answers
78 views

Prove that the cardinality of $\{0,1\}^R$ is not equal to that of $R$

From what I understood $\{0,1\}^R$ is the set of all functions from $\{0,1\}$ to $R$. I would be happy not only for the proof but a good and maybe simplified explanation of the concept of Aleph's and ...
1
vote
2answers
76 views

Confusion regarding one formulation of the Axiom of Choice.

One formulation of the Axiom of Choice is: The Cartesian product of non-empty sets is always non-empty. Cartesian product is defined as making "every possible pair" between elements of two sets. ...
0
votes
2answers
51 views

can you define this set?

I have troubles finding a set of numbers which are in the set $ M \subset \mathbb R^2$ which is made up of $(x,y)\in\mathbb{N}\times\mathbb{N}$, which satisfy the conditions $ x\leq 4$ and $y\leq 4$. ...
0
votes
3answers
55 views

Are these sets countable or uncountable?

I get that to prove a set is countable you have to find a bijection from that set to a countable set, but i usually get stuck trying to find that function. I'm trying to prove whether or not these ...
1
vote
2answers
74 views

Are There Numbers Beyond the Reals?

I understand that reals are defined as "completing" the rationals, which (at least in ZFC) are in turn derived from the natural numbers. So, if ordinal numbers are viewed as an extension of the ...
1
vote
1answer
18 views

Definition of some sets

I need to know what the sets $\mathbb{Z}^{[0,1]}$, $[0,1]^{\mathbb{Z}}$ and $\mathbb{Z}^{\mathbb{Z}}$ are. So, could someone tell their definition. Thanks.