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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2
votes
3answers
35 views

Let $A$ be a subset of a topological space. Prove that $Cl(A) = Int(A) \cup Bd(A)$

Let $A$ be a subset of a topological space. Prove that $Cl(A) = Int(A) \cup Bd(A)$ Here are my defintions: Closure: Let $(X,\mathfrak T)$ be a topological space and let $ A \subseteq X$ . The ...
1
vote
1answer
45 views

Show that the following map is a bijection

$$g(x,y) = \frac{1}{2}(x-1)x + y$$ where $g:\mathbb{Z}^+ \times \mathbb{Z}^+ \longrightarrow \mathbb{Z}^+$ Attempt: I am only having problems with proving the injectivity part so that is all I'm ...
1
vote
1answer
34 views

Let $A = [0,1) \cup (3,4]$ be a subset of $(\mathbb R, \mathfrak T_H)$

Let $A = [0,1) \cup (3,4]$ be a subset of $(\mathbb R, \mathfrak T_H)$ $\mathfrak T_H$ is the collection of all subsets of $U$ of $\mathbb R$ such that either $U = \emptyset$ or for each $x \in U$ ...
1
vote
1answer
42 views

Let $f:\mathbb R \rightarrow \mathbb R$ be given by $ f(x) = x^2 -3$

Let $f:\mathbb R \rightarrow \mathbb R$ be given by $ f(x) = x^2 -3$ Find $f([-2,1])$= $[-3,1]$ Find $f^{-1}([-2,1])$= $[-1,1]$ I am not wonderful at these types of problems and I seem to make ...
1
vote
3answers
90 views

Is $\{\varnothing\}$ a subset of $\varnothing$, the empty set?

Formed a table for a competitive exam: $\begin{array}{cc} \text{True} & \text{False} \\ \hline \varnothing\subseteq\varnothing & \varnothing \in \varnothing\\ ...
0
votes
5answers
83 views

Show that the set $A \cap B = \emptyset$

Let $A$ and $B$ be two sets for which the following applies: $A \cup B = (A \cap B^{C}) \cup (A^{C} \cap B)$. Show that $A \cap B = \emptyset$. How?! I am seriously stuck. One thought I had is to ...
1
vote
2answers
37 views

Ordered Pairs (Ordering multiple elements)

I have been doing some reading on set theory and I have come across "ordered pairs". I understand that sets in general are unordered, and when we want to place them in a particular order we use the ...
3
votes
1answer
66 views

Prove $\limsup{A_n}\backslash\liminf{A_n}=\limsup{(A_{n+1}\backslash A_n)}$

$\newcommand{\N}{\mathbb{N}}$ Problem: Prove $\limsup{A_n}\backslash\liminf{A_n}=\limsup{(A_{n+1}\backslash A_n)}$ Attempt(Revised): (I am not sure if it's correct. I would appreciate if anyone can ...
0
votes
2answers
22 views

Define $f : X \rightarrow Y$ by $f(x) = y_0$ for every $x \in X$. Then $f$ is $\mathfrak T_1 - \mathfrak T_2$ continuous.

Suppose that $(X, \mathfrak T_1)$ and $(Y, \mathfrak T_2)$ are topological spaces and suppose $y_0 \in Y$. Define $f : X \rightarrow Y$ by $f(x) = y_0$ for every $x \in X$. Then $f$ is $\mathfrak ...
1
vote
3answers
49 views

How am I to interpret this result?

Prove of give counter example: Let A and B be sets. $$ A \backslash ( A \backslash B) = B \backslash ( B \backslash A) $$ An attempt at a proof: If $ x \in A \backslash ( A \backslash B)$ then $x ...
3
votes
2answers
26 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then $A'$ is a closed set.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then $A'$ is a closed set. ($A'$ is the set of all limit points) I originally thought this was a true statement ...
0
votes
1answer
44 views

How do you evaluate $\bigcup\bigcup S$ if $S = \{\{a\},\{a,b\}\}$?

How do you evaluate $\bigcup\bigcup S$ if $S = \{\{a\},\{a,b\}\}$? I understand that $\bigcup S = \{a,b\}$ but how can I find $\bigcup\{a,b\}$ if I don't even know what the elements of $a$ and $b$ ...
1
vote
2answers
35 views

Arrangements of sets of k positions in a n-competitors race

Let $E(n)$ be the set of all possible ending arrangements of a race of $n$ competitors. Obviously, because it's a race, each one of the $n$ competitors wants to win. Hence, the order of the ...
0
votes
1answer
26 views

Check the identity $A\cap (B - C)=(A\cap B) - (A\cap C)$ [duplicate]

$$A\cap (B - C)=(A\cap B) - (A\cap C)$$ I am trying to prove this by set algebra from left to right. I can do it from right to left and also using mutual inclusions, but get stuck in $\rightarrow$. ...
6
votes
4answers
412 views

Confusion about the null (empty) set being contained in other sets

I'm having a tough time understanding how the set theory of null sets work. I have: $$ X=\emptyset,\quad\quad Y = \{\emptyset\},\quad\quad Z = \{\{\emptyset\}\}. $$ Some of my self-study exercises ...
2
votes
3answers
50 views

Why is $\mathbb T\cup\mathbb A = \mathbb Q \cup \mathbb I =\mathbb R$?

Where $\mathbb T $ is the set of transcendental numbers, and $\mathbb I $ is the set of irrational numbers and $\mathbb A $ is the set of algebraic numbers. The sets $\mathbb Q$ and $\mathbb R$ have ...
0
votes
1answer
26 views

Prove a set obtained from sequence of measurable sets is measurable

Exercise Let $(X,\Sigma,\mu)$ be a measure space and let $(A_k)_{k \in \mathbb N}$ be a sequence of measurable sets. For each $m \in \mathbb N$, we define $B_m$ as the subset of all points in $X$ ...
2
votes
3answers
31 views

Proving the combinatorial expression

Ok I've been reading in my probability book about the different methods on how to count and I'm just trying to dissect the usual combinatorial formula: $$\binom {a} {b} = \frac{a!}{b!(a-b)!}$$ ...
0
votes
2answers
24 views

How is the cartesian product $E^F$ defined?

For arbitrary sets $E, F$, how is $E^F$ defined? It seems to be the set of all maps from $F$ to $E$, i.e. $E^F := \{\phi : \phi\colon F \rightarrow E\}$? Is that right?
0
votes
1answer
45 views

Test for countability

What is the simplest way to check that a set is countable or not? With no prior experience on such questions, I want to request an answer to the following question: Prove that the set $Z^{+} * Z^{+} ...
2
votes
2answers
31 views

If $A$ is a finite set in $(\mathbb R, \mathfrak T_U)$ then $A' = \emptyset$.

If $A$ is a finite set in $(\mathbb R, \mathfrak T_U)$ then $A' = \emptyset$. My knowledge: $\mathfrak T_U$ is the usual topology $A'$ is the set of all limit points and my definition for this is: ...
1
vote
0answers
16 views

set nested element symbol

Given a nested set $S = \{\{e_{11},e_{12},\dots,e_{1n}\},\{e_{21},e_{22},\dots,e_{2n}\},\dots,\{e_{k1},e_{k2},\dots,e_{kn}\}\}$, I want to specify a set consisting of specific subelements, for example ...
3
votes
2answers
96 views

Ordinals - motivation and rigor at the same time

Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal ...
1
vote
3answers
55 views

Are cardinal numbers sets in ZFC?

Are cardinal numbers sets in ZFC, or just proper classes? If they are sets, what is their structure?
0
votes
0answers
26 views

Is an infinite set always equinumerous to either set of natural or real numbers? [duplicate]

Is an infinite set always equinumerous to either set of natural or real numbers? Is there any set "between"? Or maybe "beyond"?
1
vote
1answer
20 views

Non-ordered n-tuple?

In many mathematics texts I've seen "ordered n-tuple" appear, and in such texts, there isn't any mention of just "n-tuple". So I'm wondering: are there really cases where one writes "n-tuple" and ...
0
votes
2answers
17 views

Set inequality with functions

Let $f:A \to B$ with $X\subset A$ and $Y \subset B$. I'm trying to prove $X \subset f^{-1}(Y) \implies f(X) \subset Y$. Note that $f^{-1}(Y)$ denote the inverse image of $Y$. I've been element ...
1
vote
1answer
33 views

Let $(X, \mathfrak T)$ be a topological space and let $A$ but a subset of $X$ then $Int(Bd(A)) = \emptyset$

Let $(X, \mathfrak T)$ be a topological space and let $A$ but a subset of $X$ then $Int(Bd(A)) = \emptyset$ I need to decide if this is true or not. I have done a little research and some ...
0
votes
3answers
160 views

Understanding the use of the Cartesian Product in the proof of $|\mathbb R\times \mathbb R|=|\mathbb R|$

Where the Cartesian Product of two sets $\mathbb A$ and $\mathbb B$ is such that $\mathbb A\times \mathbb B=\{{ (a,b)|a \in \mathbb{A}, b \in \mathbb{B}\}}$ In trying to understand the proof that ...
3
votes
3answers
25 views

Let $A$ be a finite simply ordered set.

Show that $A$ has a largest element. [Hint: Proceed by induction on cardinality of $A$] Attempt: According to the assumption my set $A$ is finite and simply ordered so that would mean $A = \{A_1, ...
2
votes
3answers
61 views

The number of elements in a set

I have a small task, part of my homework, which tends to confuse me because of its simplicity. It makes me think that I am missing something. I have to find the number of elements in the set {w | w ∈ ...
1
vote
3answers
37 views

Is there anything to prove in this corollary?

Show that if $B$ is not finite and $B\subset A$, then A is not finite. I mean the statement is very trivial, but I'm having an issue actually writing what I would deem a good proof of this. The only ...
0
votes
1answer
15 views

Find the number of factors of [2^(15).3^(10).5^(6)] which are squares , cubes or both.

I can find that there are 7 squares of 2, 5 squares of 3 & 3 squares of 5.(of the given number) I can also figure out that there are 5 cubes of 2 , 3 cubes of 3 & 2 cubes of 5.(of the given ...
2
votes
2answers
36 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then 1. $Cl(A) = Cl(Int(A))$ 2. $Int(A) = Int(Cl(A))$

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then $Cl(A) = Cl(Int(A))$ $Int(A) = Int(Cl(A))$ I believe both of these statements are false and I think I ...
0
votes
1answer
27 views

What is the roster form of this set?

Set $$A = \{x:x\in \mathbb{Z}^+, x<10 \text{ and } 2^{x}-1\text{ is odd}\}$$ Shouldn't the Roster form be : $$A = \{1,2,3,4,5,6,7,8,9\}$$ ? or would it be $$A = \{(2^{1}-1), (2^{2}-1), ...
1
vote
1answer
27 views

A statement w.r.t. the injection of $\mathbb{Z^+}$

Let $v$ be an injection form $\mathbb{Z^+}$ to $\mathbb{Z^+}$ without any fixed points. Denote the image of $v$ as $v\left(\mathbb{Z^+}\right)$. Take a subset $S \in v\left(\mathbb{Z^+}\right)$ s.t. ...
0
votes
1answer
27 views

Equivalence classes of $\Bbb Z$ with the operation $\mod n$

So I came across this phrase in my abstract algebra textbook: The integers $\mod n$ also partition $\Bbb Z$ into $n$ different equivalence classes; we will denote the set of these equivalence ...
1
vote
2answers
68 views

The length of a point and the interval

I think the length of a point is $0$, and since biunique corespondence between the points of [0, 1] and [0, 10], therefore I came to the conclusion that there is a same number of points between [0, 1] ...
4
votes
3answers
52 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of X. If $A\subseteq B$ then $A' \subseteq B'$

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of X. If $A\subseteq B$ then $Bd(A) \subseteq Bd(B)$ If $A\subseteq B$ then $A' \subseteq B'$ ($A'$ is the set ...
1
vote
5answers
65 views

Suppose that $A$ and $B$ are sets and that $f: A \rightarrow B$ is onto. Does being onto guarantee the sets are finite?

Suppose that $A$ and $B$ are sets and that $f: A \rightarrow B$ is onto. Determine which of the following statements are true: If $A$ is finite then $B$ is finite. If $B$ is finite, ...
0
votes
0answers
17 views

Pairwise crossing lines meaning

I'm having a hard time being certain of the meaning "pairwise crossing" in the context of Graph Theory... namely, if say 4 lines are pairwise crossing, may any be parallel. the question states: "A ...
4
votes
0answers
41 views

finding a bijective function from the real plane to the real line

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove explicitly (don't use any theorems or known facts, but find a bijective function) that ...
1
vote
2answers
30 views

Problem involving finite sets condition

I stumbled upon this innocent looking problem in my old high school algebra textbook and I just can't figure it out . It goes like this : How many finite, non-empty sets satisfy the following ...
0
votes
0answers
22 views

finding an injective function to prove cardinality equality

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove that the set of all binary sequences (sequences of $0$ and $1$) except for the binary ...
2
votes
4answers
77 views

Proving that $A\subset B$ if given $A=A\cap B$

Let $A = A \cap B$. Prove $A \subseteq B$ I go about like this : Let $x \in (A \cup B)$ $\implies x\in A ~~\text{and} ~~ x\in B$ Question 1 : Is this true? Will and come here? Ideally or ...
0
votes
1answer
31 views

Proving that there are as many infinite binary sequences and infinite binary sequences not containing 11

I need to prove that all the infinite binary sequence are equal in cardinality to the infinite binary sequences which don't include 1 twice in a row. And I'm supposed to use ...
8
votes
5answers
466 views

Precise meaning of “extension”?

Halmos's Naive Set Theory explains the "extensionality" in "axiom of extensionality" as: Every set is determined by its extension. and that's it. What is a set's extension, then? Intuitively it ...
1
vote
2answers
23 views

Name of an element of an element of a set

Is there some way of notating that an object can be related to a set through element relations, even if it is not an element of the set? e.g. $a\not\in\{\{a,b\},\{c,d\}\}$, but ...
0
votes
0answers
21 views

Show that $(0,1]$ and $(0,1)$ are equinumerous. [duplicate]

I know that one approach is to find a bijective function and another to find a set which is equinumerous to both of them, but I dont have ideas for any of those approaches. Any suggestions?
1
vote
3answers
44 views

Show that $A \cap B = B$ iff $A \cup B = A$, where $A \subseteq B$.

Show that $A \cap B = B$ iff $A \cup B = A$, where $A \subseteq B$. I have tried to do this by element-chasing, but I just end up saying that $x \in A$ and $x \in B$. I am really stuck for ...