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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-1
votes
3answers
62 views

If $\{x\}=\{z\}$, does that necessarily mean that $x=z$?

Proposition. If $\{\{x\},\{x,y\}\}=\{\{z\},\{z,t\}\}$ then $x=z$ and $y=t$. My question is that if $\{x\}=\{z\}$ then, must $x=z$ be?
0
votes
1answer
29 views

Is there an example where $ A \subseteq\mathcal {P}\bigcup A\ $ is no longer true?

I came up with the following: Let A = {x} Then $ \bigcup A = x $ $ \mathcal {P} \bigcup A =$ ? This is where I get stuck. The definition of power set is the set of all subsets of $A$ ...
0
votes
0answers
33 views

Is $D$ a field?

Problem. Let $D$ be an integral domain and let $\mathcal{F}(D)$ be a field of quotients of $D$. If $D\subset \mathcal{F}(D)$ then prove or disprove that, $D$ is a field. ...
0
votes
2answers
43 views

Prove the existence of a bijection

Let $A$ and $B$ be sets, and suppose $A$ is infinite. Let $B$ be a countably infinite subset of $A$. Show that if $f: \mathbb{N} \to B$ and $g: B \to \mathbb{N}$ are bijections, then $$ h: A \to A- ...
0
votes
0answers
12 views

Let $A$ be a nonempty set. Prove that if $S$ and $R$ are equivalence relations on $A$, then $S \cup R$ is both reflexive and symmetric.

Let $A$ be a nonempty set. Prove that if $S$ and $R$ are equivalence relations on $A$, then $S \cup R$ is both reflexive and symmetric. My method: Let $x \in A$ be given. Then $x \in S$ or $x \in ...
1
vote
1answer
33 views

Injective, Surjective, and Cardinality

Let $f:(0,\infty)\to (0,1)$ given by $f(x) = \frac{x}{x+1}$. Decide whether $f$ is injective and whether it is surjective? What does this say about the cardinality of $R$ and $(0,1)$? I am not how to ...
0
votes
1answer
33 views

Is $f$ surjective and injective?

Let $f$ be a function from $f:\mathbb{N}\to \mathbb{Z}$ given by $f(a) = (-1)^aa$. Decide whether $f$ is injective and whether it is surjective. Which function do I start with in determining this? ...
0
votes
1answer
27 views

Verify the following statement, $ \{\{x\}, \{x,y\}\} \in\ A \implies \{x,y\} \in\bigcup A $?

My attempt at solving it: Let $ A = \{a,b,c,d\} $ where: $ a = \{x\} , b = \{x,y\} , c = \{x,y,z\} $ and $ d = \{\{x\}, \{x,y\}\} $ Then, $ \bigcup A =a \cup\ b \cup\ c \cup\ d = ...
0
votes
1answer
24 views

Open Interval in subsets

Suppose that $X$ is a subset of $\Bbb R$. If there exists an open interval contained in $X$, then $\# X = \# \Bbb R$.
0
votes
4answers
75 views

Equivalence relation for “almost equal to” [closed]

Let $A$ and $B$ be subsets of the set of natural numbers. We say that $A$ is almost equal to $B$, and write $A \approx B$, if there exists a finite subset $X \subset$ (natural numbers) such that $A ...
1
vote
1answer
15 views

Prove that if $f : A \rightarrow B$ is a function, $D \subseteq A$, and $E \subseteq A$ then $f(D) - f(E) \subseteq f(D - E)$.

Prove that if $f : A \rightarrow B$ is a function, $D \subseteq A$, and $E \subseteq A$ then $f(D) - f(E) \subseteq f(D - E)$. My method: Let $y \in f(D) - f(E)$. Hence $y \in f(D)$ and $y \notin ...
-3
votes
1answer
74 views

$\mathbb N\times\mathbb N$ is countable

$$\mathbb N\times\mathbb N \text{ is countable}.$$ Is there any way to prove it using induction? without fundamental theorem of arithmetic
1
vote
1answer
20 views

Complement of the universal set (U) is the empty set (∅)

The question ask me to state True or False and give reasons. However I prefer True. Reason: The Universal set denoted by (U) is simply a set of all given sets and complement is simply saying that ...
2
votes
0answers
45 views

$\mathrm{R}^n$ for non-integer values of $n$?

If I have a set of all possible $x$ for $x \in \mathrm{R}^n$ for non-natural number $n$, what meaning would this have and how would it be used? For instance, if I had $\mathrm{R}^{2.5}$, would this ...
-6
votes
1answer
48 views

How many sets can we create?

If a set is defined based on the following three points: 1)It has exactly three elements; 2)All elements are in Arithmetic Progression; 3)All elements are primes. Eg: $\{3,5,7\}$. How many such ...
1
vote
1answer
22 views

is it true that $A \cap (E_1^c \backslash E_2^c) \subset E_2$

For $A,E_1,E_2 \in \mathcal{F}$ where $\mathcal{F}$ is a $\sigma-$algebra is it true that: $$A \cap (E_1^c \backslash E_2^c) \subset E_2?$$
0
votes
1answer
20 views

Power set intersection proof

In book "Naive Set Theory" I am asked to prove that intersection of power sets is equal to power set of intersection $P(E) \cap P(F) = P(E \cap F)$ I am not sure if I got definition of power sets ...
-4
votes
1answer
80 views

Prove that set {1, 0} exists.

I have an exercise: Prove that set {1, 0} exists. Set theory is quite confusing for me to grasp at the moment but I know few axioms that should prove that this is set but how do I actually show ...
1
vote
4answers
61 views

Under certain conditions $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}\Rightarrow \{a,b,c\}=\{a',b',c'\}$

Let $a,b,c,a',b',c'\in \mathbb{Z}_{\geq 1}$ be such that $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1,\quad \frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}<1. $$ Suppose $$ ...
0
votes
1answer
14 views

Unique combinations of the elements of two sets

What will be the mathematical notation for a set of playing cards obtained in such a way, so the cards from all ranks $R = \{Ace, 2, 3, ..., King\}$ are part of it and a card of a particular rank may ...
1
vote
2answers
52 views

If $A,B$ are sets and $B$ is finite, and there is an injection $f:A \to B$, then $A$ is finite and $card(A) \leq card(B)$

I am stuck with proving this statement for a while. Prop. If $A,B$ are sets and $B$ is finite, and there is an injection $f:A \to B$, then $A$ is finite and $\operatorname{card}(A) \leq ...
1
vote
1answer
40 views

Proving the Axiom of Choice is equalivalent to the statement “If $A$ can be well-ordered, then so can $\mathcal{P}(A)$.”

I am still not completely confident in proving equivalence between the Axiom of Choice and statements such as the one posed in the title, so I want to make sure that I am on the right track. So ...
1
vote
1answer
28 views

Why is this set club?

I am currently reading a proof on properties of stationary sets and one step of the proof does not make a whole lot of sense to me. The proof asserts that If $\kappa$ is a regular cardinal and ...
0
votes
0answers
17 views

Let $S$ be an infinite set. Then there is a set $A\subset S$ such that $A\sim \mathbb{N}$. [duplicate]

This proof seems to follow very quickly from the definition of infinite sets, but I feel like what I have is incomplete: Since $S$ is infinite, we have that $S\sim \mathbb{N}$. By definition, $S$ is ...
1
vote
1answer
42 views

How has the teaching of (undergraduate) Set Theory changed over time?

I'm writing an Essay on Set Theory, and realized it was formulated quite recently, so I thought it might be cool to have some first person accounts. Russell's Paradox was discovered just around a ...
1
vote
2answers
155 views

Schröder-Bernstein theorem and injective maps

I am a bit confused about the statement of the Schröder-Bernstein theorem which states the following: Suppose that $A$ and $B$ are sets, and that $f : A \to B$ and $g : B \to A$ are injective ...
2
votes
2answers
30 views

Does every metric on a non empty set can be extended on a super set to a metric?

Let $\phi \ne X \subseteq Y$ , let $d$ be a metric on $X$ , then does there exist a metric $d'$ on $Y$ such that $d(x,y)=d'(x,y) , \forall x, y \in X$ ? What if we also assume that the metric $d$ on ...
0
votes
1answer
20 views

constructing a disjoint set

Let $F_1,F_2,\dots$ be sets in some sigma algebra, let $E_i = F_i \setminus\bigcup_{j<i} F_j$. It is true that now $E_i$ and $F_i$ are disjoint, but is it true that $\bigcup_{i=1}^n E_i = ...
2
votes
1answer
50 views

prove C is a proper subset of ℕ. Then prove ℕ is infinite.

Please, can you help me to do this? Let $C =\{n + n \;|\; n\in \mathbb{N} \}$, and define $f:\mathbb{N}\to C$ by $f(n) = n + n$. First prove $C$ is a proper subset of $\mathbb{N}$. Then prove ...
0
votes
1answer
44 views

prove: ℕ is countably infinite

Please, I need someone to help me to prove this theorem ℕ is countably infinite. I know how can I prove it when the set define by something, but I'm confused and don't know how do this. Thanks
0
votes
1answer
37 views

List the partitions of the set $S = \{1, 2, 3\}$.

I write the partition sets of set $S=\{1,2,3\}$ as follows: $\{\{1, 2, 3\}, \{1\}, \{2\}, \{3\}\}$ can someone show me why and how to complete the list?
0
votes
2answers
45 views

Confused with logic

I have taken a picture of the solution to the proof of the following question. Let $X$ be a topological space. A family of subsets $(V_i)_{i \in I}$ of $X$ is said to have the finite interseciton ...
1
vote
1answer
29 views

Understanding a detail about functions

Let $\sigma, \tau, \pi$ be functions such that the following compositions make sense. Assume the following is true: $\sigma$ is surjective(not sure this is needed here), and for $\sigma$ and $\tau$ ...
12
votes
1answer
270 views

How to formulate the P v.s. NP problem as a formal statement inside the language of set theory?

I've read a lot that some computer scientists believe that P v.s. NP could turn out to be independent of ZFC. The thing that puzzled me is how to formulate this inside the language of set theory? I ...
-3
votes
1answer
40 views

Is this subset of the unit cube compact? [closed]

Given $x_1,x_2 \in \{y \in \mathbb{R}^n: y\ge0, 1^Tx=1\}$. I have the set $S=\{0 \le q\le 1, x_1^Tq \le c\}$ where $c \in [0,1]$ and the inequalities are to be understood component-wise. The set ...
0
votes
3answers
56 views

Prove that if A is an infinite set and $|A| =|B|$, then $B$ is an infinite set.

Prove that if $A$ is an infinite set and $|A| = |B|$, then B is an infinite set. Proof: Suppose $A$ is an infinite set and $|A| =|B|$, and that $B$ is a finite set. Because $B$ is a finite set, ...
-1
votes
1answer
41 views

$\mathbb{R}^2$ to $\mathbb{R}^1$ Injective Mapping While Preserving the Triangle Inequality

Is there a way to map from $\mathbb{R}^2$ to $\mathbb{R}^1$, such that every point in $\mathbb{R}^2$ has a unique point in $\mathbb{R}^1$ and you preserve the distance (isometry) relations of ...
1
vote
0answers
27 views

Are $\mathbb{R}^1$ and $\mathbb{R}^2$ the same infinity? [duplicate]

I've seen a proof that shows that rational numbers are not the same infinity as real numbers. I'm curious if there is a proof that shows $\mathbb{R}^1$ and $\mathbb{R}^2$ are the same infinity. If ...
4
votes
1answer
26 views

Methods to find $f$, given the functions $f \circ g$ and $g$

There is one way, which is to use the fact that $ \ f(g(g^{-1}(x)))=f(x)$. But this method only works if $g$ has a right inverse. There are other heuristic methods, which is to "guess the shape" of $ ...
0
votes
0answers
57 views

Example of a collection that is not a set?

Just out of curiosity (the subject is way out of my league at the moment) I have been reading a little about set theory, and I came across Russel's paradox. From what I understood, Russel's paradox ...
1
vote
1answer
19 views

For $A \subseteq \mathbb R$, $\exists \Sigma$, s.t. $A \notin \Sigma$?

For any set $A \subseteq \mathbb R$, there exists a sigma algebra $\Sigma$ of subsets of $\mathbb R$ such that $A \notin \Sigma$. Is this true or false? I would think false because we can easily ...
0
votes
0answers
25 views

problem involving distributive union over intersections of sets

I am trying to solve this problem but I can't find the key to it (I'm still beginner in set theory): Let $A_{1},...., A_{n}$ be arbitrary events, and put $U_{k} = \bigcup (A_{i_{1}} \bigcap ... ...
2
votes
2answers
34 views

Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$?

If I use the above statement, provided that it is right, in a question, would I have to prove it as well?
0
votes
1answer
37 views

Express a relation as a function $f: A \to \mathcal{P} (A)$

Explicitly give the "$\leq$" relation shown in the following graph as a function from $A$ to $\mathcal{P} (A)$. I think this is an odd question as I don't think this is even technically a ...
3
votes
3answers
70 views

If $f$ function then $f^{-1}$ function iff $f$ function injective (one-to-one).

During the lecture we learned this phrase: "If $f$ is a function then $f^{-1}$ is a function iff $f$ is injective (one-to-one)." But why? What with onto? $f$ doesn't need to be Surjective ...
0
votes
1answer
17 views

What does it mean for transitive closure to be minimal?

I don't understand what that word "minimal" has to do with anything. Can anyone explain and give an example?
0
votes
2answers
25 views

Infinite Sets Proof - Integer Sets

Let $Z^- $ be the set of negative numbers. Prove $Z^-$ ≈ $Z^+$ by finding a bijective function $f : Z^+-> Z^+$. Prove that the function is bijective. Could someone tell me how to get started on ...
2
votes
4answers
70 views

Prove that if $A$, $B$, and $C$ are sets then $(A - B) \cup (A - C) = A - (B \cap C)$ [duplicate]

Prove that if $A$, $B$, and $C$ are sets then $(A - B) \cup (A - C) = A - (B \cap C)$. I have the proof for the first direction: Let $x \in (A - B) \cup (A - C)$ be given. Hence, $x \in (A - B)$ ...
0
votes
0answers
47 views

Closed set minus an open set.

If $A= [0,1]\cup[2,3]\cup \{4\}$ and $B= (0,1)\cup(2,3)$ What would $A\setminus B$ be? I believe the answer would be $\{4\}$ but I think I may be wrong....
0
votes
1answer
20 views

Basic set notation in combining different ranges of numbers

What is the proper way to specify a set which contains all even numbers between 1 and 10, and all odd numbers between 11 and 30? Would this work? $$ U = \{n, m\ |\ n \ \text{is even},\ 1 \le n \le ...