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, ...

learn more… | top users | synonyms

0
votes
2answers
72 views

Equality in set theory

In Introduction to Axiomatic Set Theory by G. Takeuti and W. M. Zaring chapter 3 It is given: Definition of equality as: $a=b \Leftrightarrow (\forall x)[x \in a \Leftrightarrow x \in b]$. And it ...
-3
votes
4answers
44 views

Find an explicit bijection from the Natural numbers to any finite subset of the Natural numbers [on hold]

Yeah, it's a question for my topology class and I'm completely stuck. Edit: Sorry about the confusion: Find an explicit bijection (one-to-one, onto function) between N and the set of all finite ...
0
votes
0answers
12 views

Order on the set of partitions (terminology)

Let $S$ and $T$ be partitions of some set $U$. What is the name for the partition $\{ X\cap Y \mid X\in S, Y\in T, X\cap Y\ne\emptyset \}$? Should it be called the infimum of $S$ and $T$? meet of ...
-1
votes
0answers
30 views

Applications of infinite cardinalities in real analysis

What are some topics in real analysis that make use of infinite cardinalities larger than that of the real numbers themselves, preferably topics that are widely applied in scientific applications?
0
votes
1answer
48 views

At Most Countable Sets: Finite vs Countable

Quick question: I understand that finite sets are equivalent to $J_n$ for some n $\in$ N, and that countable sets are equivalent to N. Also, either of these is true if and only if an injective map f ...
1
vote
2answers
20 views

Proving a bijection(injection and surjection) over a function

I need some help proving bijections: Suppose f is a function from $$ \mathbb R^2 \rightarrow \mathbb R^2$$ Defined by $$f(x,y) = (ax-by,bx+ay)$$ Where a,b are numbers with $$ a^2 + b^2 \neq 0 $$ ...
-4
votes
4answers
59 views

Is ∅ equivalent to {∅}?

Is ∅ equivalent to {∅}? I think they are, but I am not sure? If anyone could clarify, that would be great. Thank you!
0
votes
2answers
16 views

Proofing sets and subsets

If S ∩ T = S, then S ⊆ T. I have no idea where to start What I have done so far: Suppose S and T are two sets and assume the fact that S ∩ T = S Let (x ∈ S ∩ T), then: (x ∈ S) Λ (x ∈ T) ≡ (x ∈ S) ...
0
votes
0answers
30 views

Proof of the uncountability of reals using the diagonal argument—problem?

Consider a common proof of the uncountability of $(0,1]$, as presented here for example: We assume that the reals can be arranged in a sequence $x_k$, represent every number in $x_k$ by its ...
1
vote
1answer
25 views

Sets Theory Disproof

I have to disprove the statement: For all sets $S$, if $S$ is a subset of the Natural Numbers, then there must exists some $t ∈ S$ such that $|t|\ge1$ Any hints?
0
votes
1answer
25 views

Find a 1-1 function mapping the interval (0,1) to the set of rational numbers

Written another way, $f : (0,1) \to \mathbb{R}$ where $f$ is a bijection. I can't think of a function capable of this. If I just map the input to the same output value the function will never reach ...
1
vote
3answers
44 views

Does the set given by $\{(1/n)\}_{n=1}^\infty$ include $0$?

Is there some sort of consensus on whether or not $$0 \in \{(1/n)\}_{n=1}^\infty?$$
-1
votes
1answer
35 views

How do I create an injection here? [duplicate]

I am trying to show that $|\Bbb {R} \times \Bbb {R}| \leq |\Bbb {R}|$. I don't know how to define $f:\Bbb {R} \times\Bbb {R} \rightarrow \Bbb {R}$ in a way that would make $f$ injective. My ...
-1
votes
0answers
30 views

Is $^\mathbb{N}\mathbb{R}$ $\sim$ $^\mathbb{R}\mathbb{N}$? [duplicate]

Is $^\mathbb{N}\mathbb{R}$ $\sim$ $^\mathbb{R}\mathbb{N}$? I know you have to use Cantor-Bernstein, and prove both directions, but i don't know how to start the proof
-1
votes
1answer
23 views

Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R?

Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R? I don't even know how to start. I know to be a equivalence relation R must be reflexive, symmetric and ...
1
vote
2answers
36 views

If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$

If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$. [Hint: Given $g$ mapping $B$ onto $A$, let $f(X)=g^{-1}(X)$ for all $X \subseteq A$] I follow the hint and obtain the function $f$. ...
0
votes
1answer
18 views

Cardinality of sets regarding

Consider the following sets of functions on $\mathbb{R}$. $W=$The set of all constant functions on $\mathbb{R}$ $X=$The set of polynomial functions on $\mathbb{R}$ $Y=$ The set of continuous ...
4
votes
1answer
236 views

Finding intersecting subsets for given binomial coefficient

My apologies if this question is more appropriate for mathisfun.com, but I can only get so far reading about combinatrics and set theory before the interlocking logic becomes totally blurred. If this ...
-2
votes
3answers
34 views

Cardinality of the power set

Show that the cardinality of the power set of a finite non-empty $N$ set is a multiple of $2$. Then, show that it is exactly expressed by $2^n$, where $n$ is the cardinality of $N$ and that this ...
0
votes
1answer
17 views

Cardinal numbers of Setminus

Could you help me check the following fact ? Let A,B,C be sets such that C⊆A and C⊆B. Then |A∖C|=|A|−|C| |A∖C|=|B∖C| if and only if |A|=|B| where |A| is the cardinal number of A and A∖C is ...
0
votes
0answers
12 views

Transfinite recursion and sequence function of type W into X

Transfinite Recursion Theorem: If W is well ordered set and f is sequence function of type W in a set X, then there exist unique function U from W into X such that U(a) = f(U'a') for each a in W. A ...
1
vote
1answer
15 views

Order type relation in poset and well ordered sets

I just read the definition: Two partial ordered sets X and Y are said to be similar iff there a bijective function from X to Y such that for f(x) < f(y) to occur a necessary and sufficient ...
0
votes
2answers
22 views

(Different Approach) [EDITED: Please Review] An at most countable union of at most countable sets is at most countable

Question Re-phrased: I'm having a lot of trouble wrapping my head around this problem. While I've looked through similar posts, It's difficult understanding the maths because I currently have ...
11
votes
4answers
2k views

Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
2
votes
2answers
155 views

Prove $f(S \cap T) \subseteq f(S) \cap f(T)$

$f(S \cap T) \subseteq f(S) \cap f(T)$ x lies in ($S \cap T$), which means the domain has fewer elements than the domain of S and T, since x must be in S and T. All f(x) values of x, which resides in ...
1
vote
1answer
33 views

How to prove this? “For all sets A,B⊆D and functions f:D→R, we have f(A∩B)⊆(f(A)∩f(B)).” [duplicate]

Here's my attempt: f(A∩B) = f({x|x∈A∧x∈B}) = {f(x)|x∈{x|x∈A∧x∈B}} f(A)∩f(B) = f({x|x∈A}) ∩ f({x|x∈B}) = {f(x)|x∈{x|x∈A}} ∩ {f(x)|x∈{x|x∈B}} = {x|x∈{f(x)|x∈{x|x∈A}}∧x∈{f(x)|x∈{x|x∈B}}} And now I'm ...
1
vote
1answer
24 views

A collection of pairwise disjoint open intervals must be countable

Let $U$ be a collection of pairwise disjoint open intervals. That is, members of $U$ are open intervals in $\mathbb{R}$ and any two distinct members of $U$ are disjoint. Show that $U$ is countable. ...
0
votes
1answer
20 views

Set Theory and finite unions

Let $A$ be the collection of finite unions of sets of the form $(a,b]\cap Q$ where $-\infty\leq a<b\leq \infty$. Does $\phi\in A$?
-3
votes
0answers
37 views

How a mathematician would call a set that acts like a texture in infinite surface covered with it? [on hold]

So we have a set of values (that contain as much information as one $2d$ texture tile and acts as if it is was rendered on infinite plane). So say we had a set of such type with elements $\{1, 2, 3, ...
-4
votes
1answer
46 views

How to show that $A-(B\cap \overline {C})\subseteq A\cup (B\cap C)$ [on hold]

How do I solve this? Let A, B and C be sets. Show that $A-(B\cap \overline {C})\subseteq A\cup (B\cap C)$ I can't figure it out. Could I use venn or karnaugh?
0
votes
0answers
14 views

Using the negation of a statement to disprove original statement

Prove the following statement is false by first writing the negation, then proving the negation is true: For all sets, S, if S ⊆ ℕ, then there exists some t ∈ S such that |t| ≥ 1. So far, I've ...
-1
votes
1answer
57 views

A and B are sets. Prove that if $A \subseteq B$, then $\bigcup A \subseteq \bigcup B$

A and B are sets. Prove that if $A \subseteq B$, then $\bigcup A \subseteq \bigcup B$ Here's what I have so far: Let $x\in\bigcup A=\{x\mid\exists X\in A:x\in X\}$. Therefore $x\in X$. Since ...
0
votes
1answer
55 views

Notation question in elementary set theory - what is $\bigcup A$?

Let $A$ be a set. What is defined as $\bigcup A$? Is it the union of all sets that $A$ includes? Could someone provide an example for this notation? Thank you!
1
vote
2answers
55 views

What does the notation $\bigcup_{n\in\mathbb N} A_n$ mean in sets?

$$\bigcup\limits_{n\in\mathbb N} A_n$$ The book is asking me to prove that $f(\bigcup\limits_{n\in\mathbb N} A_n) = \bigcup\limits_{n\in\mathbb N} A_n$. I'm able to prove that f(the notation ...
1
vote
1answer
844 views

Is the intersection of the function of two sets a subset of the function of the intersection of two sets?

Let X and Y be sets and let f: X --> Y be a function from X to Y. If A and B are subsets of X, is it true that f(A) intersect f(b) is a subset of f(A intersect B)? If so, prove your answer; ...
-2
votes
0answers
40 views

Let A and B be sets. Prove that if A ⊆ B, then ∪A ⊆ ∪B. [on hold]

Let A and B be sets. Prove that if A ⊆ B, then ∪A ⊆ ∪B. Note: A detailed proof please, thank you.
0
votes
1answer
28 views

What is the relation between ∪A and A?

What I mean by this is ∪A⊆A, is A⊆∪A, or is ∪A=A? I'll give an example: Let A be a set and A={B, C}, where B and C are sets. Now let's say B={1,2} and C={2,3}. This means A={{1,2},{2,3}} and ...
3
votes
1answer
223 views

If x is an element of y and y is an element of z, is x an element of z?

Let x∈y and y∈z. Does this imply that x∈z? For example: Let y={A,B} and z={{A,B},C}. If x=A, then x∈y. My understanding, however, is that x is not an element of z since A is not an element of z.
2
votes
2answers
28 views

Can a binary relation on a set $S$ isomorphically embed every binary relation on $S$?

Is there any binary relation $R$ on a non-empty set $S$ such that $R$ isomorphically embeds every binary relation on $S$? (By "$R$ isomorphically embeds $Q$" I mean: there is a one-to-one function ...
-2
votes
0answers
17 views

What kind of sets are described by $\{x: x \in S_i$ for at least one $i\}$ and $\{x: x \in S_i$ for every $i\}$. [on hold]

What kind of sets are described by $\{x: x \in S_i$ for at least one $i\}$ and $\{x: x \in S_i$ for every $i\}$> Specifically, how does one express these sets in non -set-builder notation? Here, ...
-1
votes
5answers
80 views

Why the term “countable”?

In my computer science theory class, we are discussing the concept of countability. I understand the concept, but the choice to use the word countability seems absolutely unintuitive to me. Why was ...
0
votes
1answer
34 views

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring sets?

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring on sets? By a closed-open interval , I mean an interval of the form $[x,y)$ A ring of sets is a non-empty ...
-1
votes
3answers
43 views

Prove that if A is a subset of B, then union A is a subset of union B. [on hold]

Let A and B be sets. Prove that if A ⊆ B, then ∪A ⊆ ∪B.
1
vote
1answer
33 views

Equivalence between two topological statements concerning the basis of a topology.

I need to show the following statement Let $\mathcal{B}\subset P(X)$ be a set of subsets of a set $X$, such that $\bigcup_{U\in \mathcal{B}}U =X$ then the following are equivalent $i)$ there ...
0
votes
1answer
27 views

Is there a powerset equivalent to the Kleene star?

For some arbitrary alphabet E, is there an equivalent way to construct E* using powersets, sets, or sequences?
17
votes
7answers
2k views

There is no smallest infinity in calculus?

Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess. From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and ...
0
votes
0answers
38 views

Proving cardinality of the reals and the cross product of the reals [duplicate]

I am trying to prove that $\Bbb {R} \times \Bbb {R} \sim \Bbb {R}$ using the Cantor-Bernstein Theorem. So then that would mean that I need to prove that $|\Bbb {R}| \leq |\Bbb {R} \times \Bbb {R}|$ ...
0
votes
1answer
19 views

Double Complement of a set proof

Question states: Prove the law of double complements for sets: If $A$ is a set and $A^\complement$ is its complement than prove that: $$ (A^\complement)^\complement = A$$ I started with: $$ ...
0
votes
0answers
13 views

Extension of Premeasures

Here, a premeasure is a countably additive set function whereas a measure is one acting on a sigma-algebra. Not every positive premeasure admits an extension to a positive measure as the following ...
1
vote
2answers
20 views

Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a finite measure space $\mu(\Omega)<\infty$. It is well known that a measure is continuous from above as ...