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

0
votes
0answers
10 views

Simple question: How can you make an infinite subset over a finite set

My early morning brain is not computing this possibility. Let $\sigma$ be a finite set, how can $L \subset \sigma$ be made into a infinite subset?
0
votes
1answer
36 views

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

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 ...
0
votes
1answer
22 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
0answers
16 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
23 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 ...
1
vote
1answer
30 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 ...
2
votes
1answer
26 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 ...
5
votes
3answers
39 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 $2^4-1$ ...
10
votes
0answers
123 views

What does it take to divide by $2$ (or even $3$)?

Theorem 1 [ZFC, classical logic]: If $A,B$ are sets such that $\textbf{2}\times A\cong \textbf{2}\times B$, then $A\cong B$. That's because the axiom of choice allows for the definition of ...
0
votes
1answer
17 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 ...
1
vote
1answer
22 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, ...
1
vote
2answers
2k views

Maximal and Minimal Elements

In my textbook, the give an example for finding maximal and minimal elements on a set. The set is $(\{2,4,5,10,12,20,25\},|)$. To find the maximal and minimal elements of the set, the draw a Hasse ...
0
votes
0answers
23 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) ...
0
votes
5answers
56 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 ...
-1
votes
2answers
39 views

Basic set theory proof [on hold]

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.
3
votes
2answers
41 views
+50

Is there a specific name for this set of square-rooted primes?

Consider the set of all the primes numbers (± square rooted) and all the irrational numbers that can be formed under their addition (only the addition of finitely many elements is allowed, i.e. no ...
5
votes
0answers
45 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 ...
0
votes
1answer
49 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
2
votes
1answer
28 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
51 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
34 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
52 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
31 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
18 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
31 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: $$\rho : P(\Bbb{N} \times \Bbb{N}) \rightarrow P(\Bbb{N} \times \Bbb{N})$$ $$\rho(r) = \bigcup\{ r^{2^n}|n\in\Bbb{N}\}$$ ...
19
votes
10answers
3k views

Defeating Russell's paradox

I am not very big in mathematics yet(will be hopefully), naive set theory has a problem with Russell's paradox, how do they defeat this sort of problem in mathematics? Is there a greater form of set ...
1
vote
2answers
27 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
18 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
26 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 ? ...
6
votes
0answers
85 views

Is there really anything wrong with Bourbaki's Set Theory?

Recently I have started reading Bourbaki's Theory of Sets on my own. Regarding one of the explanations of a concept when I went to a Professor of our college, he asked me why I was wasting my time ...
0
votes
1answer
33 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
0answers
35 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 ...
0
votes
1answer
26 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 ...
2
votes
1answer
38 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? $$(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$$
1
vote
3answers
120 views

Munkres Chapter 1 Section 7 Exercise 8

Let $X$ denote the two element set $\{0,1\}$; let $X^\omega$ denote the set of all the binary sequences; and let $B$ denote the set of countable subsets of $X^\omega$. Then how to see if $X^\omega$ ...
1
vote
0answers
100 views

Borel Measures: Atoms (Summary)

Disclaimer: The question here has been solved, now: Finest Measurable Partition (For jeapardy it is stated below, anyway. Have fun! ;) ) Summary: This is a summary of the discussions: ...
0
votes
0answers
60 views
+200

Cardinal number of the set of all one-to-one mappings of $A$ onto itself.

This is an exercise in Naive Set Theory by P. R. Halmos. If $\text{card }A=a$, what is the cardinal number of the set of all one-to-one mappings of $A$ onto itself? What is the cardinal number ...
4
votes
2answers
250 views

What does the completed graph of a function mean

zab said: the Levy metric between two distribution functions $F$ and $G$ is simply the Hausdorff distance $d_C$ between the closures of the completed graphs of $F$ and $G$. I have difficulty in ...
0
votes
2answers
33 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 ...
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, ...
0
votes
0answers
23 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 ...
2
votes
2answers
38 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 ...
1
vote
1answer
22 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 ...
5
votes
2answers
179 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 ...
-2
votes
1answer
32 views

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

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
1answer
18 views

inductively defined group statements

If A is a set, and $B_{1}, B_{2}\subseteq A$ subsets of $A$. Also, $f_{1}:A\rightarrow A$ and $f_{2}:A\rightarrow A$ we will mark: $F1={f1} , F2={f2}$ How do I prove the following: $X_{B1\cap ...
1
vote
5answers
276 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 ...
0
votes
0answers
17 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 ...
0
votes
2answers
17 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 ...