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, ...

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0
votes
2answers
27 views

The cardinalities [duplicate]

I have some difficulties to resolve a problem. Could you explain me why this sets: $[a,b]=[c,d]$,where $a,b,c,d \in\mathbb{R}$ with $a<b$ and $c<d$ have the same cardinalities? thanks
3
votes
1answer
50 views

Is the following a semiring?

I have the following problem: Let $f: X' \rightarrow X$ be any map and $\mathcal{H} \subseteq \mathcal{P}(X)$ a semring. Is $f^{-1}(\mathcal{H})$ a semiring? Thanks for your help!
0
votes
1answer
70 views

Geometrical description of cantor set is uncountable

Why is Cantor set uncountable? I would like to intuitively understand the uncountable nature of Cantor set. When I construct Cantor set I do not feel so.
0
votes
1answer
34 views

Prove the following $\neg((A\cap B)\cup (\neg A \cap C)) = (A\cap\neg B)\cup (\neg A\cap \neg C)$ [duplicate]

How can I prove the following statements are equivalent using laws of set theory? $\neg((A\cap B)\cup (\neg A \cap C)) = (A\cap\neg B)\cup (\neg A\cap \neg C)$ Using De Morgans laws to simplify the ...
0
votes
4answers
43 views

Can someone explain the meaning of \ in operations with sets? [duplicate]

I have never faced with such operator... what does '\' mean? Does this expression make any sense? (A ∪ B) \ C = A ∪ (B \ C)
1
vote
1answer
50 views

How to prove that $\neg((A\cap B)\cup (\neg A \cap C)) = (A\cap\neg B)\cup (\neg A\cap \neg C)$

How can I prove the following statements are equivalent using laws of set theory? $\neg((A\cap B)\cup (\neg A \cap C)) = (A\cap\neg B)\cup (\neg A\cap \neg C)$ I managed to use De Morgans laws to ...
0
votes
3answers
64 views

Prove that a set is countable..

Iam supposed to prove that the set $\mathbb{Z}_+ \times \mathbb{Z}_+$ is countable by first constructing a bijective function $f: \mathbb{Z}_+ \times \mathbb{Z}_+ \to A$ where $A$ is the subset of ...
-2
votes
2answers
63 views

What is the power of the set of $A :=\{\varnothing , \{\varnothing\}\}$.

a) How many elements does the set $\{\varnothing, \{\varnothing,\{\varnothing\}\}, \{\{\varnothing\}\}, \{\{\varnothing\},\varnothing,\varnothing\}\}$ have? b) What is the power of the set of $A ...
2
votes
3answers
97 views

$\mathbf{Q}$ basis of $\mathbf{R}$.

Could someone give me an explicit basis of $\mathbf{R}$ as a vector space over $\mathbf{Q}$? I no some linearly independent subset, namely $1,e,e^2,\ldots$ but this seems to be a deep result already ...
3
votes
1answer
42 views

Recommendation on setting the reference axis for mathematical objects

(I don't know what the title should be for this post, please change it if you have a better title. Also tags) In many situations, there arises cases that one mathematical structure embeds into ...
4
votes
4answers
88 views

Prove that $\text{int(intA)=int(A)}$?

I want to prove that $\text{int(intA)=int(A)}$ (and we are in metric space). I have two questions regarding this. (1). I came up with this proof but don't know if it's correct or not. First I use ...
0
votes
1answer
53 views

Showing a bijection without a specific function (and only with cardinality)

$\forall n \in \mathbb{N}$, define $I_n = \{k \in \mathbb{N} \mid k \leq n\}$. And for any set $A$, the number of elements in $A$ is $n$ if there exists a bijection from $I_n$ to $A$. Say the ...
0
votes
2answers
42 views

Correct notation for union of all elements in a set?

Say I have a set $H$ and I want to describe the union of all elements in $H$. How would I write that? I believe I've seen a big U with a subscript used before.
0
votes
3answers
73 views

How do I proof that the set of finite subsets of $\mathbb{N}$ is countable? [duplicate]

I am working myself into analysis 1 and i came across countable and uncountable sets. Problem 1: I am very confused about the two terms and I would be very thankful if somebody could explain the ...
3
votes
1answer
73 views

Cardinality of basis of endormophism algebra

Is there a relation between the cardinality of the basis of a vector space $V$ over $k$ and the cardinality of the basis of $\operatorname{End}{V}$, the set of $k$-linear endormophisms of $V$, over ...
1
vote
2answers
59 views

Cardinality of $R^n$ and $R$ is equal [duplicate]

How can we prove that cardinality of set of all real numbers $R$ and $R^n$ is equal for every $n$ in the set of all natural numbers?
-1
votes
2answers
60 views

Prove that $(A \cup B)−(A \cap B) = (A− B) \cup (B − A)$

Prove that $$(A \cup B)−(A \cap B) = (A− B) \cup (B − A)$$ by showing that the left hand side is the subset of the right hand side and vice versa. Progress Let $x$ be an element of LHS. Then x is ...
1
vote
1answer
52 views

Some problems from section 4 of Munkres

I'm right now covering Section 4 of Topology by James R. Munkres, 2nd edition, and am stuck with the following problems in the exercise set after Section 4: Problem 8(c): Show that given $a$ with ...
0
votes
2answers
65 views

Proof of $(A \cap B) \cup (B \cap C) \cup (C \cap A) = (A \cup B) \cap (B \cup C) \cap (C \cup A)$

Prove that $(A \cap B) \cup (B \cap C) \cup (C \cap A) = (A \cup B) \cap (B \cup C) \cap (C \cup A)$ I have noted $\mathcal{A} = x \in A$, $\mathcal{B} = x \in B$, $\mathcal{C} = x \in C$ So in ...
1
vote
0answers
19 views

Let $B$ and $C$ be sets. Suppose $|B| = m$ and $|C| = n$. Find the following cardinality:

Let $B$ and $C$ be sets. Suppose $|B| = m$ and $|C| = n$. Find the cardinality: $|\mathcal{P}(B \times \mathcal{P}(C))|$ I want to check if the following is correct: The cardinality of ...
0
votes
1answer
26 views

Show power set union identity

Show if $a \in X$ then $P(a) \in P(P(\cup X))$ I define $P(a)= \left\{a:a \subseteq X\right\} $ I also have the identity $P(S) \cup P(T) \subseteq P(S\cup T)$, but I'm not sure how to start the ...
4
votes
6answers
566 views

What is an example of function f: N to Z that is a bijection?

Could you give me an example of function $ f \colon \mathbb N \to \mathbb Z$ that is both one-to-one and onto? Does this work: $f(n) := n \times (-1)^n$? $\mathbb N$ starts with zero.
0
votes
4answers
63 views

A intersection (A union B)

I'm trying to prove $A \cap (A \cup B) = A$. I'm stuck on the last part of my proof, not sure how to show next: $$x \in A \cap (A \cup B)$$ $$\iff x \in A \;\;\text{and}\;\; x \in A \cup B$$ $$\iff x ...
3
votes
2answers
36 views

Exercise on posets and antichains in Steven Roman's Lattices and Ordered Sets

I have just began reading through Steven Roman's "Lattices and ordered sets", and I came across an exercise in Chapter 1 that I can't seem to find a good answer to. All the others are fairly easy, so ...
0
votes
1answer
28 views

Proving $(X \triangle Z) \backslash (Y \triangle Z) \subseteq (X \backslash Y) \triangle Z$

How can I prove $(X \triangle Z) \backslash (Y \triangle Z) \subseteq (X \backslash Y) \triangle Z$? I have tried looking what means $x \in (X \triangle Z) \backslash (Y \triangle Z),$ but I got to a ...
0
votes
1answer
21 views

Proof using the symmetric difference

I just started my Analysis course, and I have little idea about the method for proving these things. The symmetric difference of two sets $X,Y$ is defined as $$X\Delta Y=(X\setminus Y)\cup(Y\setminus ...
1
vote
1answer
24 views

Help with excercise about symmetric difference

I am asked to prove if any of the sets $(X \backslash Y) \triangle Z$ and $(X \backslash Z) \triangle (Y \backslash Z)$ is a subset of the other. I am trying to get some understanding of these sets by ...
0
votes
1answer
25 views

A question about indexed intersections and unions

I want to show that $X\cup \underset{i}{\large\cap} Y_i=\underset{i}{\large\cap}(X\cup Y_i)$ This is what I got, but I just want to know if I am going about this the correct way. $\begin{equation} ...
1
vote
2answers
87 views

Algebraic Structures: Does Order Matter?

(I want to link to similar question with a very good answer: Question about Algebraic structure?) An algebraic structure is an ordered tuple of sets. One of these is called the underlying set, and ...
0
votes
0answers
21 views

Is the Cardinality of the Set of Contingent Propositions the Same as the Cardinality of the Set of Tautologies?

By a "contingent proposition" I mean a proposition which is neither a tautology or contradiction. Or in other words, there exists at least one valuation of the variables such that the formula ...
0
votes
1answer
184 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
2
votes
1answer
34 views

Proving that $\bigcup\mathcal{A}\times\bigcap\mathcal{B}\subseteq\bigcup\{a\times b\mid a\in\mathcal{A}\land b\in\mathcal{B}\}$ is always true

The problem is to prove that the following expression is true for any families of sets $\mathcal{A}$ and $\mathcal{B}$ that are not empty. ...
0
votes
1answer
30 views

Common Binary Relation that is not a partial order

What is an example of a common binary relation on a set A that would not be partially ordered? The most common binary relations I've seen are $=, <, \leq$, but all of these are partial orders. ...
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votes
0answers
34 views

Prove or disprove conjecture [duplicate]

Prove or disprove conjecture $$A \cap (B \cup C)=(A \cap B) \cup (A \cap C)$$ I don't know how to answer or where to start
-1
votes
3answers
41 views

Enumeration of a set

What do this statement mean: For any given $k\in \left[a,b\right]$, let $\left( x_n \right)_{n=1}^{\infty}$ be an enumeration of the set $\left\{c_jy_k^i:i,j\ge 1, \right\}$ where $c_j$ is a real ...
1
vote
2answers
43 views

What is the smallest infinite ordinal that is not order isomorphic to a reordering of the natural numbers?

I've been working on a particular set theory problem for a while and essentially I've hit a roadblock because of this question. I just need to know what this ordinal is, because I have a sneaking ...
2
votes
3answers
130 views

intro level set theory question.

I want to show that $X\cap(Y\cup Z) = (X\cap Y) \cup Z \ \iff Z\subseteq X$ I think I have figured out one direction: Assume $Z\subseteq X$ then $Z\cap X= Z$. We know $X\cap(Y\cup Z)=(X\cap Y)\cup ...
0
votes
0answers
12 views

If $V$ is a subspace of $\mathbb{R^n}$ and $V_B = \{[\vec{v}]_B : \vec{v} \in V\}$, $V_B$ is also a subspace of $\mathbb{R^n}$

Consider a basis $B$ of $\mathbb{R^n}$, $V$ a subspace of $\mathbb{R^n}$, and $V_B = \{[\vec{v}]_B : \vec{v} \in V\}$. Show that $V_B$ is also a subspace of $\mathbb{R^n}$ and dim$(V)$ = dim$(V_B)$. ...
1
vote
0answers
27 views

bijective function with powerset and symmetric difference

Let $X$ a set and $Y\subseteq X$ and $$f: \mathcal{P}(X)\rightarrow \mathcal{P}(X),\text{ with }A\in \mathcal{P}(X)\mapsto A\,\Delta\, Y$$ ($\Delta:=$ exclusive disjunction) Show that $f$ is ...
2
votes
2answers
34 views

2 element subsets of n elements?

the question is as follows: Give a recursive definition for the number of $2$-element subsets of $n$ elements. We started working this out in class and here is where we got too: -if $n = 0$, then ...
0
votes
1answer
58 views

Physical significance of the fact that the cardinality of the real number line is the same as a finite interval of the real number line

It is known that the cardinality of the real number line is the same as a finite interval of the real number line. Is there a physical meaning of this apparently conter-intuitive statement?
3
votes
3answers
118 views

Need a possible proof for $A\cap B=A\cup B$

I need to prove that- If $A,B\subseteq U$ where $U$ is the universal set, then $A\cap B=A\cup B$. I've been thinking a lot about it and I don;t think it's possible. Yet, I would like to confirm it ...
2
votes
3answers
37 views

The set of subsets of size $k$ in $\mathbb{N}$ is countable

Let $P_{k}(\mathbb{N}) = \{ A \subset \mathbb{N} \ | \ |A|=k \}$. I want to prove that $P_k$ is countable for each $k$. So I showed that this was a set of countable subsets, but I am not sure how to ...
0
votes
0answers
11 views

Term of partially ordered set with “levels”

Suppose that we have a partially ordered set $(X,\leq)$ such that the following condition holds: There exists a disjoint partition $X = \bigcup_{ i \in \mathbb N_0 } X_i$ such that for $i < j$ we ...
0
votes
1answer
18 views

Sets with all or none of the elements also being subsets - Cohn - Classic Algebra Page 11

Give examples of sets such that $(i)$ all and $(ii)$ none of their members are also subsets Firstly I should make sure I understand this correctly: The subsets of the set $S=\{\emptyset,1,2,3\}$ ...
1
vote
3answers
44 views

Union of two points

So the union of two sets is just a set that included all the elements from them but what happens for two points? For example, the set $X = [1,2] \cup [3,4]$. What would be considered in this set?
2
votes
2answers
23 views

Write out the following set by listing its elements between braces.

Write out the following set by listing its elements between braces: $\begin{align} \{X \subseteq \mathbb{N}: |X| \leq 1 \} &=\{\emptyset,\{1\},\{2\},\{3\},\ldots\}\end{align} $ Is my answer ...
6
votes
3answers
412 views

Is O(n) a proper class or a set?

Is $O(n)$ as the collection of all functions that are bounded above by $n$ a proper class or just a set? What about $O(\infty)$?
6
votes
0answers
57 views

Name of a certain set

I want to know if there is any already-standard way to refer to the set described as follows. Take the set of all primes in $\mathbb{Z}$, call it $\mathbb{P}$. Take the set of all finite products of ...
0
votes
1answer
34 views

Properties of the relation {<1,0>} on the empty set??

Suppose you take the relation R containing just the ordered pair <1,0>. Can this be a relation on the empty set? One line of reasoning might be that R is an equivalence relation on the empty set, ...