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, ...

learn more… | top users | synonyms

0
votes
1answer
23 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
37 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
19 views

Find transitive closure of $D_r$

This is one of the problem I have been solving in Velleman's How to prove book: Find the reflexive, symmetric and transitive closures of the following ...
1
vote
1answer
14 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
0answers
21 views

Prove or disprove: For all sets A and B, AI (AUB) = A

I understand basic set theory, but I am not able to wrap my head around this one. The lecture did not cover this topic and I cannot find info online. What exactly is meany by "AI." In other words, ...
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
57 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
17 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
114 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
40 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 ...
0
votes
3answers
23 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
73 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
154 views

Set Theory and $1 = 0.999\dots$ [on hold]

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
32 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
76 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
44 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?
0
votes
0answers
37 views

Of which sets does the $\sigma$-algebra generated by the first $n$ one-point sets of $\mathbb{N}$ consist?

Let $n \in \mathbb{N}$ and $\mathcal{E}_n := \{\{1\},\{2\},\dots,\{n\}\}$. The $\sigma$-algebra which is generated by $\mathcal{E}_n$ is defined as follows: $$\sigma(\mathcal{E}_n) := \bigcap ...
0
votes
1answer
23 views

Can I have a critique of this set theory proof/Advice on a similar proof?

This is an exercise from Mendelson's Introduction to Topology. The first part is to prove, given a function $\ f:A \rightarrow B$, that $\ X \subset f^{-1}(f(X))$ for all $\ X \subset A$. Here's my ...
16
votes
5answers
1k views

Naive Set Theory by Halmos is confusing to a layman like me

I want to be able to express set notations fluently in math fields used in machine learning, so I started reading Naive Set Theory by Halmos. But I have been facing a lot of problems like : On ...
0
votes
3answers
82 views

Proving $a + a = a$ if a statement is true

That's the question : Let $a$ be a cardinality such that this following statment is true : For every $A, C$, if $ A \subseteq C$, $|A| = a$ and $|C| > a$, then $|C \setminus A| > |A|$. ...
-4
votes
2answers
40 views

No. of surjections [on hold]

Find the number of surjections from a $3$-element set to a $2$-element set. Find a formula for the number of surjections from $ℙ_{k+1}→ℙ_k.$ Find a formula for the number of surjections from ...
0
votes
0answers
15 views

Support of Distribution Function

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
1
vote
3answers
42 views

Verifying $(A \bigtriangleup B) \cup C=(A \cup C) \bigtriangleup (B\setminus C)$

I am trying to do self-study out of a set theory book. In one of the question sections, it asks to verify the following identity: $$(A \bigtriangleup B) \cup C=(A \cup C) \bigtriangleup (B\setminus ...
0
votes
0answers
12 views

Closure, Interior, Boundary and Exterior of a Set in different topologies…

Closure, Interior, Boundary and Exterior of a Set in different topologies... This is something I seem to be majorly struggling with I am looking at the set of all $A = \mathbb R - \mathbb Q $. I ...
-3
votes
1answer
24 views

What is mutually disjoint sets

What is mutually disjoint sets? I know it has something to do with subsets but I don't know for sure.
0
votes
1answer
14 views

Saying that a set is a subset of random elements from another set?

For context, I'm looking to prove one of DeMorgan's Laws (I just started reading AFCLA; I'm on the section where set notation is being introduced). The one I'm trying to prove, in particular, is $$ ...
0
votes
2answers
21 views

What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them

As the title suggests, the question is : What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them ? I'll tell you where im stuck, let's say f is ...
1
vote
0answers
12 views

How to go about proving non enumerability.

I'm learning about enumerability and I'm working through an exercise which states: Show that the set of satisfiable $S_{\infty} sentences$ is not R-enumerable. I've been told to look at these ...
2
votes
1answer
31 views

Proving that if $S$ has an infinite subset then $S$ is infinite

Definition$\quad$ A set $S$ can be defined as infinite if there exists a mapping from $S$ to $S$ that is one-to-one but not onto. Otherwise, $S$ is finite. Problem: Using the definition of ...
0
votes
1answer
23 views

Bounded Set for Rational Number

$$A = \{x \in \Bbb Q: x^2 < 11\}$$ I am looking for the upper and lower bounds of set $A$. I know they should be the greatest and the smallest rational number within $(- \sqrt{11}, \sqrt{11})$ ...
1
vote
2answers
39 views

Power sets and functions

Let $a\colon\mathcal P(\mathbb N)\to\mathbb N$ be the function defined by $a(X)$ equals $0$ if $X$ has infinitely many elements and $a(X)$ equals the number of elements in $X$ if $X$ has finitely many ...
3
votes
1answer
23 views

Elementary question on set theory

Suppose $A \subset B$ then does this imply $B^{c} \subset A^{c}$? Here, $B^{c}$ denotes the complement of $B$. I have tried drawing Venn Diagrams and it seems obvious but is there a formal rigorous ...
0
votes
2answers
41 views

Having trouble understanding enumerability

From my book it states that a set is enumerable if there is a procedure which eventually yields as outputs exactly the words in the set. I can see that an example of enumerability is the set of prime ...
0
votes
2answers
62 views

Bijection between $\mathbb{Z} \longmapsto \mathbb{R}$ [duplicate]

I recently learned of Cantor's diagonal argument, and was thinking about why there can't be a bijection between any infinite set of integers and any infinite set of real numbers. I understood the ...
-4
votes
0answers
23 views

For any sets A and B, if P(A)∪P(B)=P(A∪B)then eitherA⊆B or B⊆A [duplicate]

Pls help me out with the proof:For any sets A and B, if P(A)∪P(B)=P(A∪B)then eitherA⊆B or B⊆A.
0
votes
1answer
15 views

Make the set $R$ transitive

Let $R$ be a relation on a set $A=\{w,x,y,z\}$ defined by $R=\{(w,x),(y,x),(x,y),(z,z)\}$. Using the original relation, $R$, make the necessary minimal additions to make $R$ transitive. I thought ...
0
votes
1answer
24 views

Finding an Injection

I need to prove the set A={1/n: n$\in$$\mathbb{Z}\backslash${0}} is countably infinite. To prove it is infinite, I said consider the set B={1/n: n$\in$$\mathbb{Z}^+$}, and note that B$\subseteq$A. ...
0
votes
2answers
28 views

Let $ X = \{a, b, c\}$ and $\mathfrak T = \{X, \emptyset, \{a,b\}, \{a\}\}$ Let $ A = \{a,c\}$

Let $ X = \{a, b, c\}$ and $\mathfrak T = \{X, \emptyset, \{a,b\}, \{a\}\}$ Let $ A = \{a,c\}$ I have to find each of the following sets and I think I am on the right track but I know I am not ...
0
votes
1answer
31 views

Set Theory/Intuitive Set Theory

Let $A_1, A_2, ..., A_i$ be sets, and define $S_{x}$ to be $A_{1}\cup A_{2}\cup ... \cup A_{x}$ for $x=1, 2, 3, ..., r$. Show that $\alpha=\{A_{1}, (A_{2}-S_{1}), ..., (A_{r}-S_{r-1})\}$ is a ...
0
votes
1answer
14 views

σ -algebra by choosing sets, where either the set or its complement is countable: is the complement countable?

I am reading Schilling's “Measures, integrals and martingales”, where on page 15 he constructs a $σ$-algebra, according to: $$ \mathcal{A} = \lbrace A \subset X: \# A \leq \# \mathbb{N} \quad ...