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
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3answers
30 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
44 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
27 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
14 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 ...
2
votes
2answers
72 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
52 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
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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
17 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
15 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
30 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
156 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
24 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
54 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
13 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
33 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
19 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
26 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
25 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
65 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] ...
3
votes
3answers
48 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
61 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
15 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
36 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
29 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
21 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
75 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
29 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
454 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
22 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
43 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 ...
1
vote
3answers
70 views

Show that ~ creates a partition of $M_2(\mathbb{R})$

Let $M_2 (\mathbb{R})$ be the set of 2x2 matrices over $\mathbb{R}$: $$ M_2 (\mathbb{R}) = \biggl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \; \biggm| \; \text{with } a,b,c,d \in ...
1
vote
1answer
19 views

What is the notation for a sequence of elements (non number elements)?

I am new to math and am exploring how to formally represent a sequence of events. I want to be able to say "an event sequence $E_s$ is a sequence of events $\langle e_1, e_2,\ldots, e_n\rangle$". Just ...
1
vote
1answer
25 views

Let $A = (0,1) \cup [2,3)$ be a subset of $(\mathbb R, \mathfrak T_C)$. Find the following sets. How am I doing?

Let $A = (0,1) \cup [2,3)$ be a subset of $(\mathbb R, \mathfrak T_C)$ $\mathfrak T_C = \{(a,\infty) :a \in \mathbb R\} \cup \{\mathbb R, \emptyset\}$ I need to find the following sets: Int(A) ...
1
vote
3answers
36 views

Give a family of sets $\{F_a\}$ with each $F_a\subseteq (0,1)$ and $F_a \cap F_b \neq \emptyset$ but $\bigcap_aF_a = \emptyset$.

Give a family of sets $\{F_a\}$ with each $F_a\subseteq (0,1)$ and $F_a \cap F_b \neq \emptyset$ but $\bigcap_aF_a = \emptyset$. What is an example of such a family of sets?
3
votes
1answer
44 views

Is this proof that $\kappa^{<\kappa}=\kappa$, when $2^{<\kappa}=\kappa$, correct?

Let $\kappa$ be a cardinal number. I want to show that if $\kappa$ is regular and if $2^{<\kappa} = \kappa$ then $\kappa^{<\kappa}= \kappa$. Here is what I got so far: $$ ...
-1
votes
2answers
35 views

Proof of surjectivity of a function [closed]

Let $g:[0,\infty) \rightarrow [0,1)$ defined by $g(x)= \frac{x}{1+x}$. How i can prove that's onto, with the definition?
0
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2answers
40 views

Is there a sample of a $f(x)=y$ multivalued function whose inverse $f(y)=x$ is also multivalued?

Trying to learn about the properties of the multivalued functions, I found the definition at the Wikipedia as "a left-total relation (that is, every input is associated with at least one output) in ...
5
votes
2answers
79 views

When can variables simply be variables?

This may seem a somewhat strange question, but I've been tying myself in knots about it recently. When constructing a polynomial ring, you must formally define a polynomial as an ordered ω-tuple, ...
3
votes
2answers
64 views

Prove every subset of $\Bbb N$ is countable.

This isn't a homework problem. I've seen a proof of the following statement online, and I think the proof is suspect, or at least incomplete. Theorem. Every subset of $\Bbb N$ is countable. ...
0
votes
2answers
36 views

no. of elements in $(B \times A) \cap (A \times B)$

Can someone prove that $n((A \times B) \cap (B \times A)) = n(A \cap B)^2$ I am unable to derive this. (Frankly, I don't know how to even start). I just let $(A \times B) = M, and (B \times A) = N$, ...
1
vote
1answer
23 views

intersection closure for boolean functions

This seems a basic thing yet I'm having hard time understanding it. Let $X=\{x_1,x_2,\dots,x_n\}$ be a set of $n$ elements and $S$ be a set of boolean functions from $X$ to $\{0,1\}$. So every ...
2
votes
1answer
64 views

Can we prove AC from the statement “There is no $\aleph$ cardinal strictly between $\operatorname{CARD}(X)$ and $\operatorname{CARD}(2^X)$”?

If $X$ is a set, let $\operatorname{CARD}(X)$ denote the Cardinal number of $X$. Let GCH(1) be the statement "If $K$ is an infinite initial ordinal number, then there exists no initial ordinal number ...
0
votes
1answer
32 views

Show $f^{-1}(\bigcup_{B_i\in B}B_i) = \bigcup f^{-1}(B_i)$ proof done correct?

Show $f^{-1}(\bigcup_{B_i\in B}B_i) = \bigcup f^{-1}(B_i)$ Attmept: I used induction: We know it is true for $i = 1$ where $i\in \mathbb N$ Confirm for $i = 2$: $f^{-1}(B_1\cup B_2) = f^{-1}(B_1) ...
0
votes
0answers
21 views

Let $(X, \mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) \cap Cl(X-A)$.

Let $(X, \mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) \cap Cl(X-A)$. I know this is a true statement. I am trying to prove if because I would also ...
1
vote
5answers
67 views

what does function from a set to its power set mean?

I am having some confusion in understanding, what exactly does a function from a set to a power set means. I don't want a proof to the cantor's theorem. Consider a set A = {1,2,3} , P(A) = ...
0
votes
1answer
27 views

De Morgan's laws [duplicate]

one can prove de Morgan's laws in set theory by induction. Are these laws true for an uncountable set of indices ( sets ) ? Can somebody give a proof for one of the laws in the case of uncountable ...
1
vote
5answers
189 views

Is this element a subset of set A? [duplicate]

Let $A$ be the set $\{\propto,\{1,\propto\},\{3\},\{\{1,3\}\},3\}$. The statement "$\{1,\propto\}\subseteq A$" is false. (Taken from A Concise Introduction to Pure Mathematics) But I think ...