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, ...

learn more… | top users | synonyms

3
votes
2answers
386 views

What is the meaning and difference of { } and ( )

I know that $(1,2,3) \ne (3,2,1)$ but {1,2,3} = {3,2,1}. But what does it mean actually? And maybe some explanation on why 1. Ø subset of {0} is true? 2. {x} element of {x} is false?
0
votes
2answers
38 views

Subsets of a power set.

How many subsets T of the power set of A contain at most 2 elements if the cardinality of A is n where n is a natural number ? The number of elements in power set A = 2^n. But i dont know how to ...
1
vote
0answers
51 views

Set and category theory

I have difficulties with the notions of set and category theory. I don't understand the difference between: a set, a collection and a universe! Although I understand why the collection of all sets ...
1
vote
3answers
32 views

Determining the list elements of $U = \{(A,B)\in \mathcal P(X) ×\mathcal P(X)\mid A=(X−B)\}$

Define $X = \{1,2,3,\ldots,n\}$, for some positive integer $n$. The set $U$, is defined as: $U =\{(A,B)\in \mathcal P(X) ×\mathcal P (X)\mid A=(X−B)\}$. If $n=3$, show the elements of $U$. I ...
0
votes
1answer
22 views

Proof Help for $[(A\cap B = B \cap C)\wedge (A\cup B = B \cup C)] \implies A = B$

$[(A\cap B = B \cap C)\wedge (A\cup B = B \cup C)] \implies A = B$ We must show that our conditions show that $\forall x[x\in A \rightarrow x\in B \wedge x\in B \rightarrow x \in A]$ $x \in A ...
0
votes
2answers
23 views

Composite relations defined by xRy

Let R be the relation defined on $\mathbb{Z}^{+}$ defined by $xRy$ if and only if $x < y$. $R$ can be written as $\{(x, y) | y − x > 0\}$. How do I find the composite relation $R^n$ for $n \in ...
5
votes
2answers
82 views

Prove that if $A$ is an infinite set then $A \times 2$ is equipotent to $A$

I want to prove that if $A$ in an infinite set, then the cartesian product of $A$ with 2 (the set whose only elements are 0 and 1) is equipotent to $A$. I'm allowed to use Zorn's Lemma, but I can't ...
3
votes
0answers
39 views

Topological proof for this set theory statement

Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite. I already found a boring brute ...
2
votes
2answers
85 views

A question on empty set and Russell's paradox

Suppose $S$ is a well-defined set and $A$ is meant to be a subset of $S$ that is defined as follows: $A = \{x|(x\in S)\wedge(x\not\in S)\}$. Is $A$ the empty set $\varnothing$, since it is based on ...
1
vote
1answer
30 views

sequence of close and bounded sets in a prefect space

Suppose that$(E_n)$$_{n \in \mathbb N}$ be a sequence of closed and bounded sets in complete space $M$ such that $ E_{n+1} \subseteq E_n$ for all $ n \in\mathbb N$. If $\lim \operatorname{diam} E_n ...
1
vote
3answers
40 views

Is the empty set the only possible set for $A$ such that $A=\{x|x\not\in A\}$?

Is the empty set the only possible set for $A$ such that $A=\{x|x\not\in A\}$?
0
votes
1answer
26 views

The relation of domain and image of a function and its inverse

Theorem: Let both $f$ and $f^{-1}$ be functions. $\newcommand{\dom}{\operatorname{dom}}\newcommand{\im}{\operatorname{im}}$ Then $\dom(f) = \im(f^{-1})$ and $\dom(f^{-1}) = \im(f)$. Let $f: X ...
1
vote
2answers
46 views

Uncountably infinite set between 1-2 and 1-10?

Just a quick question: Is the size of the set of real numbers from 1 to 2 greater, or equal in size to the number of real numbers between 1 and 10? I'm a Physicist so I'm not totally clued up on ...
0
votes
1answer
37 views

Is the relation on integers, defined by $(a,b)\in R\iff a=5q+b$, a function? [closed]

Let $A=B=\mathbb N$. Relation $R$ is: $(a,b)\in R$ iff for some $q \in \mathbb Z$ we have $a=5q+b$ Given a relation, show that it's a function. To Show: $\forall a \in A \ \exists b \in B$ such ...
5
votes
3answers
52 views

Subsets of $\{1,2,3,4,5,6,7,8\}$ with at least 1 odd and 1 even number

How can I formally write the number of subsets of $S=\{1,2,3,4,5,6,7,8\}$ with at least 1 odd and 1 even number? I know if I take the subset with even numbers, $E =\{2,4,6,8\}$, there are $2^4-1$ ...
1
vote
0answers
19 views

Inner Measure: Measurability

Reference This problem grew out from: Inner Measure vs. Outer Measure Setting Given a plain space $\Omega$ and a possibly empty semiring $\mathcal{S}$. Consider a premeasure ...
0
votes
1answer
33 views

Image, preimage and set operation in mappings

Not the best title but I don't know how to better describe it. So the image of a set is usually written as $f(B)$, my question is, can I use sets in the place of variables in the expression of my ...
0
votes
1answer
23 views

find image and inverse image of function

I have function $f:R\to R^2 , \ \ f(x)=<\cos 3x, \sin 3x>$ and I have to find image on the interval $(0, \pi]$ and inverse image $[0, +\infty) \times[0, +\infty)$ I think the image will be ...
0
votes
0answers
27 views

Finding a equinumerous set.

Let $A$ and $B$ any sets. How can I show that there are $A^{'} \approx A$ and $B^{'} \approx B$ such that $A^{'}\cap B^{'}=\emptyset$? ($X \approx Y \Leftrightarrow$ $X$ and $Y$ are equinumerous) ...
1
vote
5answers
59 views

Composition of two functions is not commutative

I have been always shown that the composition of two functions is, in general, not commutative with a counterexample. But can you give a more general proof of this statement (that is to say, one that ...
7
votes
1answer
144 views

Is there a linear order with this property

I was trying and failing to construct a linear order L each of whose uncountable subsets contains an uncountable well ordered subset but L is not a countable union of well ordered subsets. Is this ...
-1
votes
2answers
44 views

Basic set theory proof [closed]

How to prove $(A \times B = B \times A) \Leftrightarrow (A = \varnothing \ or \ B = \varnothing \ or \ B = A)$ ? I'm not sure about my solution.
2
votes
1answer
29 views

Find a bijection, check if a given set is a function

I have problems with two exercises: $1)$ Find a bijection between $A$ and $B$. $$A=[0,1) \times[0,1)$$ $$B=\{{<x,y>}\in \mathbb R^2: x,y>0,\ x+y<1\}$$ $2)$ Decide if the given set is a ...
1
vote
1answer
30 views

Premeasures: Inner Measure vs. Outer Measure

Problem Given a plain space $\Omega$ and a ring $\mathcal{R}$. (In fact, a semiring would do the job, too.) Consider a premeasure $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. For simplicity, ...
2
votes
1answer
29 views

$\langle \mathfrak{c},\mathfrak{c}\rangle$-Independent Matrix

Given cardinals $\lambda,\kappa$, an $\langle \lambda,\kappa\rangle$-independent matrix on $X$ is a colection $\mathcal{A} = \{A_{\alpha}^{\beta}:\alpha<\lambda\wedge \beta<\kappa\}$ sattisfying ...
0
votes
2answers
54 views

“Subset of above not equal to” $ \subsetneqq $ Symbol

I was reviewing my Algebra diary, and I noticed a symbol that I was not familiar to: $ \subsetneqq $. After some reseach on the internet I eventualy found it (through UNICODE), and found that the ...
1
vote
4answers
35 views

Select one or zero elements from a set

I am far from a mathematician. Still. I want to formally express that only 0 or 1 element of a series of sets (1...n) is selectet to form a new set. Example: I have three sets $S_1 = \{1,2,3\}$, $S_2 ...
0
votes
2answers
69 views

An isomorphic map from natural numbers to positive rational numbers that preserves addition, multiplication and order

Since $\mathbb{Q}^{+}$ is countable, there is a bijection between $\mathbb{Q}^{+}$ and $\mathbb{N}$ (0 included). Then the question now is, can we go further by constructing an isomorphic map between ...
2
votes
1answer
34 views

Very basic question about set theory: unions and intersection

Let $\{ E_n \}_{n=1}^{\infty} $be a collection of countable sets and let $$ F_k = E_k \setminus ( \bigcup_{j=1}^{k-1} E_j ) $$ Then $F_k$ are pairwise disjoint and $\bigcup^{\infty} F_k = ...
0
votes
1answer
20 views

number of antisymmetric and not irreflexive relations

What is the number of relations on a n element set that are antisymmetric and not irreflexive? I have tried doing this as fallows- no of antisymmetric relations having atleast one self pair[like ...
0
votes
1answer
33 views

What is $r^n$ where $r \subseteq P(\Bbb{N} \times \Bbb{N})$ and $n$ is natural number?

I've got a set theory problem in which I examine the following function: $$\rho : P(\Bbb{N} \times \Bbb{N}) \rightarrow P(\Bbb{N} \times \Bbb{N})$$ $$\rho(r) = \bigcup\{ r^{2^n}|n\in\Bbb{N}\}$$ ...
1
vote
2answers
29 views

How to specify each digit of a real number in decimal representation in set theory?

So real numbers have decimal representations. If you want to say the $n$th digit of some real number, how do you say this formally in set theory?
1
vote
2answers
25 views

Find the image of $A=(-2,1) \times [-2,2)$ under the function $f(x,y)=x^2y$

I have function $f(x,y)=x^2y$ and I have to find image $f[A]$ where $A=(-2,1) \times [-2,2)$ we have that $-2 < x < 1$ and $-2\le y<2$ $0 \le x^2 < 4 $ I claim that the image of ...
1
vote
1answer
27 views

Prove $B−C \subseteq A'$ implies $A \cap B \subseteq C$

Prove that if $B−C \subseteq A'$ then $A \cap B \subseteq C$. Is it perfectly reasonable to show that $A \cap B \not\subset C$, (assuming $B−C \subseteq A'$ holds) leads to a contradiction ? ...
0
votes
1answer
34 views

Number of relations on a set

What is the number of relations on a $n$ element set that are antisymmetric and not symmetric? I have soved this question using the fact that 'antisymmetric and not symmetric' means asymmetric... ...
0
votes
1answer
28 views

Algebra with set notation and set properties

Suppose that $S$ and $T$ are sets with $S \cap T = \emptyset$ Let $C \subseteq S \cup T$ and let $A = C \cap S$ and $B = C \cap T$. Show that $A \subseteq S$, and $B \subseteq T$. I said, let ...
0
votes
0answers
38 views

Why does Power Set represented by $2^x$ takes only $0$ or $1$ as values for $x$

While I was studying about Function Spaces I've seen an example of Function Space from function space of Power Set that tells that it(power set) maps from $X$ to $\{0,1\}$. I couldn't get how that ...
2
votes
1answer
40 views

How would you draw $(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$?

I know it's useful to prove set equalities to make a quick sketch of the sets described. How can I draw this one? $$(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$$
0
votes
2answers
46 views

Set theory questions - Subsets from Zorich Mathematical Analysis I

I am doing a text that my big brother gave me: Mathematical Analysis I - Zorich. This stuff is pretty hard for me, since in class we don't do sets. I can see why they are true with pictures, but i ...
2
votes
2answers
40 views

very short and basic question, is $(1,1]$ empty or is it $\{1\}$

Title says it all really. I was asked what is the union and intersection of all the sets $A_n=(1/n,1]$ where $n$ is natural. Right off the bat, $A_1=(1,1]$. is this an empty set? or is it $\{1\}$. I ...
0
votes
0answers
30 views

Help with logical equivalence proof regarding a lemma for equivalence relations.

So the question is this: Suppose A is a set. Let ~ be an equivalence relation on A and let a,b be elements of A. Then Ta = Tb if and only if a ~ b. I need to prove this statement to be true. I know ...
1
vote
1answer
48 views

Are there any sets that cannot be constructed using the symbols $\{$ and $\}$?

It seems to me all sets in mathematics can be constructed via these two symbols only. For instance, the natural numbers are defined as $0 = \varnothing = \{\}, 1 = \{0\}$, $2 = \{0, 1\}, 3 = \{0, 1, ...
1
vote
1answer
25 views

Logic and set theory proof help

Question: Prove the statement below for the sets $A$,$B$, and the universal set $U$. $$A-B=A \cap B^c $$ My attempt: Converting $A-B$ to set notation: $$A-B = \left\{ {x:x \in A, x \notin ...
-2
votes
1answer
33 views

What means $A \subsetneq X$ with A ~ X? [closed]

How it is possible to have a subset A, which is $\neq$ to X and at the same time they have an equivalence relation ~? When $A \subset X$ therefore a $\in$ A is also a $\in$ X. With A ~ X ...
0
votes
0answers
18 views

How can a function with asymptotes be defined as a mapping?

A mapping takes each element of a set S and associates it with an element t in some other set T. I believe functions to be mappings. Yet we happily call such as $\frac {x^2}{x+1}$ a function, even ...
1
vote
5answers
304 views

How to prove there is no bijection between a set and its second power set?

Let $X$ be a set. How can I show that there is no bijection between $X$ and $P(P(X))$, the powerset of the powerset of $X$. I know that there is no bijection between $X$ an $P(X)$, due to Cantor's ...
5
votes
2answers
193 views

Let : $X \to Y$ be a function. Show that if $f$ is injective then $f(A \cap B) = f(A) \cap f(B)$ for sets $A \subseteq X$ and $B \subseteq X$.

Let : $X \to Y$ be a function. Show that if $f$ is injective then $f(A \cap B) = f(A) \cap f(B)$ for sets $A \subseteq X$ and $B \subseteq X$. My answer : Suppose $f$ is injective and $f(x) \in ...
0
votes
2answers
18 views

Does $A$ is equipollent to $B$ $\implies $ $P(A)$ is equipollent to $P(B)$?

Let $A, B$ be sets and let $P(A)$ and $P(B)$ denote their powersets. Suppose there is a bijection $f: A\to B$. Is there a bijection $g:P(A)\to P(B)$? I feel like it is true, but I have trouble ...
0
votes
1answer
13 views

Is a relation transitieve if and only if $R\circ R \subset R$?

Let $X,Y, Z$ be sets and call $R\subset X\times Y$ a relation from $X$ to $Y$. Let $R$ be a relation from $X$ to $Y$ and S a relation from $Y$ to $Z$. Then composition relation is given by $S\circ ...
0
votes
1answer
35 views

Counting sets and adding an element

Let $A$ be a set with $n$ elements, where $n \in \mathbb{\omega}$. Suppose $s \notin A$, prove that $A \cup \{s\}$ has $n+1$ elements. Here is what I have done so far: By induction, let $P(n):$ if ...