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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1
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3answers
33 views

Why does $x \in A \Rightarrow x \in B$ imply $A \subset B$ and not $B \subset A$?

Why does $x \in A \Rightarrow x \in B$ imply $A \subset B$ and not $B \subset A$? When we show that $x$ is in $A$, then we show it is in $B$. So why does that imply $A\subset B$? And how do ...
1
vote
3answers
30 views

rewrite in a mathematical format

I have many sets containing three values like {1,−2,-5}. I want to write in mathematical form to filter set where there are only elements with same sign and also none of them is zero like {-1,−9,-5} ...
3
votes
2answers
32 views

For each rational $q$ there is an integer $n$ such that $q+n=271$, and vice versa

For all $q \in \Bbb Q$ there exists $n \in \Bbb Z$ so that $q + n = 271$. This is true? Because both $q$ and $n$ are rational numbers and $271$ is an integer thus it's a rational number? Also, ...
1
vote
2answers
30 views

True or false: for all subsets $A$ and $B$ of $X, f(A\cup B) = f(A) \cup f(B)\,$?

Can somebody prove For all subsets A and B of X, f(AUB) = f(A) U f(B) ? I believe that it is true, and here is my proof. If somebody sees something I did wrong, can you please explain? Take any y∈F ...
1
vote
1answer
32 views

True or false: For all subsets $ A $ and $ B $ of $ X $, if $ A \subseteq B $, then $ f[A] \subseteq f[B] $.

I am trying to determine if the following is true or false: For all subsets $ A $ and $ B $ of $ X $, if $ A \subseteq B $, then $ f[A] \subseteq f[B] $. My guess is this would be true, because ...
1
vote
1answer
25 views

Equivalence of category of subsets and subobjects

I'm trying to show that the categories $\mathcal{P}(X)$ and $Sub(X)$ are equivalent. According to Steve Awodey's "Category Theory" I need to find two functors $ E: \mathcal{P}(X) \to Sub(X)$ $F: ...
1
vote
1answer
28 views

Which equivalence class represents the zero element $0_{\mathbb Q}$ in $\mathbb Q$?

The Statement of the Problem: We identify $\mathbb Q$ with the set of equivalence classes $[a,b]$, where $(a,b) \in \mathbb Z \times \mathbb N^+$ and $(a,b) \sim (a'b')$ iff $ab'=ba'$. We define ...
1
vote
1answer
16 views

Union of equivalence relations

As part of a HW assignment in the course elementary set theory, I was given the following question: Let $A$ be a set and let $T$ and $S$ be two equivalence relations on $A$. Prove: $S\circ ...
-4
votes
0answers
15 views

Find preimage of a function given that the image of the function [on hold]

I have a following question that I need help on. Let $F(s,t)=\left( s^2 \cos(t), s^2 \sin(t) \right).$ Find the $F$-preimage of $[0,1] \times [0,1].$
-3
votes
2answers
81 views

What is the definition of a finite set $S$? [on hold]

This question is intended mainly for beginners. We can say "$S$ is not infinite" or "counting elements of $S$ is a procedure that (theoretically) terminates", a little of maths appears in “$S$ is ...
0
votes
0answers
17 views

A question on product space [duplicate]

If $|X|=\mathfrak c$, then what is the cardinality of the product space $X^{\omega}$? Thanks very much.
1
vote
1answer
21 views

Composition of equivalence relations

As part of a HW assignment in the course elementary set theory, I was given the following question: Let $A$ be a set and let $T$ and $S$ be two equivalence relations on $A$. Prove: $S\circ ...
0
votes
1answer
36 views

How to express these sets mathematically?

I'm conducting an optimization where I have a set of ordered pairs, L. In my case: L = (1,2) (2,1) etc. I also have a set Wplus which is the combination of all ...
21
votes
1answer
1k views

Is the axiom of choice really all that important?

Note about duplication link: The question advantage of accepting the axiom of choice may be seen as similar to this question in many regards, but the key difference is that I am not interested in ...
0
votes
1answer
20 views

Sum-of-divisors determinant

Let $\sigma_k(n)=\sum_{d|n}d^k$ be the generalized sum-of-divisors function. Let $S_n$ be the matrix defined by $[S_n]_{ij}=\sigma_i(j)$. I read a comment somewhere that $$\det(S_n)=1!\cdot 2!\cdots ...
2
votes
5answers
208 views

Does not being a subset of a set mean that you are a subset of the complement set?

If I have three sets $A, B$ and $C$, where $ A \cup B$ is not a subset of $C$, would it be correct to say that $A\cup B$ is a subset of $C^c$? I'm working on a proof question, and I am not sure ...
2
votes
2answers
101 views

Show that if $\mathcal{P}(A) \subseteq A $ then $ \mathcal{P}(A) \in A $.

I was working on a revision worksheet and I came across this question, and I was not sure to answer it. Can anyone help me out with this? Help will be much appreciated.
0
votes
1answer
42 views

Partition on a Closed Set A= [2,3]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
2
votes
3answers
62 views

What properties does $A\to B$ satisfy under 1-1 correspondence?

A 1-1 correspondence between two sets $A$ and $B$ is a function $f\colon A \to B$ satisfying what properties? I do know that we say that two sets $A$ and $B$ are equivalent, and we write $A \sim B$ ...
0
votes
3answers
61 views

Beginner level : What is the intuitive meaning and what are the steps to prove for an injective function

For the proof that a map is into, it is convenient to use the contrapositive of the definition of one-to-one, namely: $$ \forall x,y \in X, f(x) = f(y) \rightarrow x = y. $$ where the definition of ...
6
votes
2answers
124 views

What is the meaning of $\mathbb{R}\setminus\{0\}$?

This is used in many posts related to functions and googling it doesn't help. What does this mean? $\mathbb{R}$ should stand for all Real numbers.
1
vote
0answers
12 views

Finite intersection of arbitrary union not stable for arbitrary unions

It is a set-theoretic exercise to prove that the set of arbitrary unions of finite intersections of sets is still stable under finite intersections. However it is not true that finite intersection of ...
1
vote
1answer
42 views

In set theory, what do the subscripts $A_1,A_2,\dotsc,A_n$ mean? [on hold]

How are subscripts used in set theory, for example, In set theory, what do the subscripts $A_1,A_2,\dotsc,A_n$ mean?
1
vote
4answers
29 views

Proving Two Sets are Equal - Infinite Sets - Example

Let $$A = \{x | x = 2n+1, n\in\mathbb{Z}\}$$ and $$B = \{x | x = 2m-21, m\in\mathbb{Z}\}.$$ I am trying to prove $A =B.$ I understand that I need to prove $A\subseteq B$ and $B\subseteq A$; But my ...
0
votes
1answer
19 views

What is $h^{-1}(L)$, for $L$ a regular language and $h$ a homomorphism?

Let $L = L((00 + 1)∗)$ and $h : \{a, b\}^* \to \{0, 1\}^*$ be defined by $h(a) = 01$ and $h(b) = 10$. What is $h^{−1}(L)$? In this context "$+$" means "$\cup$". So the language $L$ is all the ...
0
votes
1answer
28 views

Prove that $\prec$ is irreflexive and transitive

Note: Definitions I use (Velleman's How To Prove It) If $A$ and $B$ are sets, then we will say that $B$ dominates $A$, and write $A \precsim B$, if there is a function $f: A \rightarrow B$ ...
1
vote
2answers
38 views

Topology without tears exercises 1.2 #6 i)

Let T be a topology on a set X such that T consists of precisely four sets; that is , $T = \{X, \emptyset, A, B\}$, where $A$ and $B$ are non empty distinct proper subsets of $X$. Prove that $A$ and ...
0
votes
1answer
22 views

Find transitive closure of $D_r = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid |x - y| = r\}$

This is one of the problems I have been solving in Velleman's How to prove book: Find the reflexive, symmetric and transitive closures of the following relations: $D_r = \{(x,y) \in ...
1
vote
1answer
15 views

Use laws of the algebra of sets to show that $X' \cap Y' = (Y \cup X)'$

Use laws of the algebra of sets to show that: $X' \cap Y' = (Y \cup X)'$ Can I get some help on how to solve this? I have tried so far to use De Morgans Laws to make it go from: $X' \cap Y'$ ...
-5
votes
2answers
28 views

Finding union and intersection of sets [on hold]

Specify the following sets: 1 : $\{1,3,5,7,9\} \cap \{2,4,6,8\}$ 2 : $\{x \in N: x + x = x^2\}\cup \{x \in N: 3x= x^2\}$ 3 : Assume that $A \cap B = 13$ and $A = 17$ and $B = 19$, ...
0
votes
1answer
14 views

Let $(X, \mathfrak T)$ be a topological space and supposed that $A$ is a subset of $X$ then the Bd(A) is a closed set.

Let $(X, \mathfrak T)$ be a topological space and supposed that $A$ is a subset of $X$ then the Bd(A) is a closed set. I am in an introduction to proofs class. I have to decided if this is a true ...
1
vote
1answer
12 views

Let $A$ be a subset of $X$. Define $\mathfrak T = \{ U: A \subseteq U\} \cup \{\emptyset\}$. Then $\mathfrak T$ is a topology on $X$.

Let $A$ be a subset of $X$. Define $\mathfrak T = \{ U: A \subseteq U\} \cup \{\emptyset\}$. Then $\mathfrak T$ is a topology on $X$. I think this is a true statement and I therefore need to prove ...
2
votes
2answers
58 views

Is $\left\{0,1,2\right\}^{\mathbb{Z}^2}=\left\{\left\{0,1,2\right\}^{\mathbb{Z}}\right\}^{\mathbb{Z}}$?

I am asking myself if $\left\{0,1,2\right\}^{\mathbb{Z}^2}=\left\{\left\{0,1,2\right\}^{\mathbb{Z}}\right\}^{\mathbb{Z}}$? Elements of $\left\{0,1,2\right\}^{\mathbb{Z}^2}$ are $0,1,2$-valued ...
-1
votes
1answer
21 views

Prove that X=Y is equivalent to P(X)=P(Y),where X,Y are sets and P(X) is Set of X's all subsets.The same goes for P(Y0 [duplicate]

Well,i haven't tried anything yet,because I've got no idea how to prove it.If possible,please help me
-2
votes
0answers
24 views

To prove ($A\cup B$) $\cap C$ = $(A \cup C) \cap (B \cup C)$ [duplicate]

I have never done rigorous et theory before .How do i prove this and generalise for $A_{i}$ ,i belonging to I ($A\cup B$) $\cap C$ = $(A \cup C) \cap (B \cup C)$ Hints ? Thanks
6
votes
2answers
115 views

If $a=b$ then $a+c=b+c$? [duplicate]

A friend of mine just asked me how to prove that if $a=b$ then $a+c=b+c$, where $a,b$ and $c$ are real numbers, I'm not sure what I should answer. I have a book called introduction to logic and to the ...
0
votes
1answer
41 views

Can you construct $\mathbb{R}^1$ from $\mathbb{R}$ using the cartesian product? If not, how is $\mathbb{R}^1$ constructed?

Can you construct $\mathbb{R}^1$ from $\mathbb{R}$ using the cartesian product? If not, how is $\mathbb{R}^1$ constructed? I'm having this doubt, I really don't know how to answer this (this is not ...
1
vote
3answers
24 views

Limit Ordinals as Infinite Ordinals and other questions

I am studying set theory and I am confused in the following: Are limit ordinals the same as infinite ordinals? I would say yes since the least non-zero limit ordinal is $\omega$. Infinite limit ...
-1
votes
1answer
28 views

Covering relation over functions

F is a group that includes all functions from N to N K is relation over F. For f,g ∈ F: (f,g) ∈ K iff ∀ n∈N, f(n)≤g(n). Obviously K is Partially ordered set and not Total Order. My problem is with ...
2
votes
0answers
35 views

Proving Finiteness

For any set $X$, if $\cup X$ is finite, then $P(X)$ (power set of X) is finite. For any Transitive set $X$, if $P(X)$ is finite, then $\cup X$ is finite. I'm a little confused with these because ...
0
votes
1answer
19 views

Cartesian product of two real sets

I've two sets, Here: A=(0,5] and B=[2,4] The following product is right or wrong? AxB=[(0,2);(0,4);(5,2);(5,4)]
1
vote
1answer
33 views

Subset of an uncountable set [duplicate]

For all uncountable set, Is there an uncountable subset such that its complement is also uncountable? How can I prove this?
-4
votes
2answers
32 views

$A\setminus (B \cup C) = ( A\setminus C) \cap (A\setminus B)$ [on hold]

Can some on help me prove showing each and every step: $A\setminus (B \cup C) = ( A\setminus C) \cap (A\setminus B)$ If anyone could shed some light on this matter
5
votes
3answers
74 views

Is the set of points or the set of lines on a plane “larger”?

Is the set of points or the set of lines on a plane "larger",or there is a 1-1 correspondence between lines and points?
6
votes
6answers
174 views

Set Theory and $1 = 0.999\dots$

If we have a set like this $\{\,0.9, 0.99, 0.999, \dots \}$ Then, will the number $1$ or $0.999\dots$ be a member of this set? My notation is synonymous to the notation presented by egreg. (or ...
-3
votes
0answers
35 views

set of numbers, elementary set theory [on hold]

im very focused to a problem concerning the continuity of the set of numbers as between 2 consecutive integers there is a infinite of fractions. this is not the real question. my question is: let be ...
1
vote
1answer
21 views

Range of $n(A \Delta B)$ in sets A and B

I was trying to solve this question- "If 2 sets A and B are such that n(A) = 15 and n(B) = 25, find the no. of elements in the range of $n(A \Delta B)$.Now, this is what I did- For $n(A\Delta B)$ to ...
0
votes
3answers
77 views

Does Russel's paradox preclude us from using the power set to generate every possible set?

Suppose I have the set of all things $\{a, b, c,... \}$. It seems to me that $ \mathcal P \{a, b, c,... \} $ would be the set of all sets, which sounds like it includes the set of all sets that do ...
0
votes
3answers
45 views

Number of elements in a set.

i am just getting started with discrete mathematics and set theory and i came across this question which would seem like an elementary problem. I would appreciate any help on this : Suppose $m$ and ...
0
votes
1answer
20 views

Substraction of two sets equivalent

What is the equivalent of $A - B$ expressed using union, intersection or compliment, where $A,B$ are sets?