Questions tagged [elementary-set-theory]

For elementary questions on set theory. Topics include intersections and unions, differences and complements, De Morgan's laws, Venn diagrams, relations and countability.

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Would this be a valid set definition for $Graph(f)$?

I had this particular set in mind: $G_f=\{(x+y, y):x \in \mathbb{R}$ and $y=f(x+y)\}$ or $G_f=\{(x+y, y):x+y \in \mathbb{R}$ and $y=f(x+y)\}$ or $G_f=\{(x+y, y):x,y \in \mathbb{R}$ and $y=f(x+y)\}$ ...
webtolight's user avatar
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1 answer
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$\sigma$-algebra on $\mathbb{R}$ generated by the collection of all one-point sets.

I have seen that the $\sigma$-algebra on $\mathbb{R}$ which is generated by the collection of all one-point sets is the collection of all sets with a countable number of elements and their complements,...
Alex A.G.'s user avatar
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Why is the set $A$ not closed under countable unions?

Let $A$ be the set of all finite unions of half open intervals of the form $(a,b], a,b \in \mathbb{R}$, $(-\infty, b]$ or $(a,\infty), a \in \mathbb{R} \cup \{-\infty\}$. I am reading that this set is ...
Alex A.G.'s user avatar
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Notation for map evaluated at all elements of a set?

Consider a map $f : X \to Y$ and some given set $A \subseteq X$. I would like to introduce the notation $f(A) = \{ f(a) \vert a \in A \} \subseteq Y$, i.e. the set of output given the elements of $A$ ...
Bart Wolleswinkel's user avatar
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Set theoretic definition of terms of the untyped lambda calculus

I am trying to translate the following definition (in Agda) of intrinsically scoped terms of the untyped lambda calculus into more mathematical (in particular set theoretical) notation: ...
user11718766's user avatar
-3 votes
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37 views

How to demonstrate that $(A ∩ B)^c ⊂ (B-A^c)^c$? [closed]

How to demonstrate that $(A ∩ B)^c ⊂ (B-A^c)^c$ ? $\begin{array}{ll} x ∈ A &x\text{ is an element of set }A\\ x ∉ A &x\text{ is not an element of the set }A\\ \emptyset &\text{Empty Set}\\ ...
Olmos Ortiz Andrea Lucia's user avatar
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Does the Foundation Axiom or the Axiom of Regularity lead to paradoxes

Does the foundation axiom or the axiom of regularity lead to paradoxes? I ask because of the following: Let D be the set of ducks Let $D^C$ be the set of things that aren’t ducks $\forall x\left(x\in ...
AUTIST INC's user avatar
-2 votes
0 answers
44 views

Proof of the principle of induction [duplicate]

I will be referencing the proof I provided here. I don't understand the remark of a person, who states the incorrectness of such proof. From what I could understand, the proof is also valid not ...
Elvis's user avatar
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Simplification of $(X_1 \times X_2) \setminus ((X_1 \setminus U) \times (X_2 \setminus V))$, $X_1, X_2$ infinite and $U \subset X_1, V \subset X_2$

I am trying to derive a simplification of this set equation. I am trying to work through it "intuitively," but I am stuck and am wondering of any set theory that can help me simplify this ...
codeing_monkey's user avatar
1 vote
1 answer
176 views

Is this proof for mathematical induction valid?

Instead of beginning from the natural numbers, we first define $\mathbb R$ using field axioms. Let $\mathscr H$ be a set of subsets of $\mathbb R$ defined as follows: $$\mathscr H = \{H \subset \...
Elvis's user avatar
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2 votes
1 answer
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Set theory: how do I "distribute" overlapping sets?

Let me preface by saying that I'm a software engineer; I'm neither a mathematician nor a logician in the academic sense. So I apologize if terms / symbols are not accurate for representing sets. EDIT: ...
Matthew Dean's user avatar
7 votes
2 answers
339 views

Number of One-One Functions

This question has been asked in my exam and I have stuck. The question says:Let $S=\{1,2,3,4,5,6\}$. The number of one-one functions $f$ defined from $S$ to $P(S)$, where $P(S)$ stands for power set ...
20DPCO190 Amanul Haque's user avatar
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Is my proof that $X \setminus (Y\cup Z)=(X\setminus Y)\cap (X \setminus Z)$ valid?

Currently studying the first chapter of Munkre's Topology, and I'm enjoying time spent proving statements. I am trying to prove De Morgan's laws. I am worried that this proof is not valid, ...
Will Fitchet's user avatar
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Computing transitive closure for relation via other relation

The closure of a relation $R$ over a set $S$ is denoted $R[S]$ and calculated via $\bigcup_{i\in\mathbb{N}}Y_i$ where $Y_0=S$ and $Y_{n+1}=Y_n\cup R(Y_n)$. ($R(Y_n)$ is the image of $Y_n$ under $R$). ...
Aresiel's user avatar
1 vote
1 answer
57 views

Are $\sin{x}$ degrees and $\sin{x}$ radians considered different functions?

As I understand, functions map elements of a set to another, so they should be indifferent to such a thing as units. If $\sin{\frac{\pi}{2}}$ is defined to be 1 then it should always equal 1, no ...
cabutchei's user avatar
0 votes
1 answer
59 views

Need explanation why this Shröder Bernstein Theorem 'proof' fails

I'm not asking to give sense for famous proofs that no Axiom-of-Choice included like Königs Dedekinds ,(... etc to be discussed in proofwiki.) I need to know what is off with the following proof ...
Zuon Erut's user avatar
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1 answer
39 views

Is the relation Anti Symmetric

The relation is Antisymmetric on $\mathbb Z^2$. Is it antisymmetric on $\mathbb R^2$. $$S=\{(x,y)\mid\exists i\in\mathbb Z,y=x^i\}\text{ on }\mathbb R^2$$ The question is about there exist any $i\in\...
Swadhin Ranjan Patra's user avatar
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1 answer
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unsure of my solution to Tao Analysis I 4th ed 3.5.2 (axiomatic set construction, cartesian products, power sets)

I am a self-teaching beginner and unfamiliar with proofs - in particular I find proposals that are intuitively true harder to prove as my brain assumes too much or skips steps. I'd like help with my ...
Penelope's user avatar
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If $x$ belongs to every set that $y$ belongs to, is $x$ equal to $y$?

This is a strengthening of my previous question, here: How to prove that if two sets belong to the same sets, then they are identical?. My current question is this. Let $x$ and $y$ be sets, and ...
user107952's user avatar
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Calculating how many classes a relation partitions a set

Just struggling to understand how to calculate how many classes a relation partitions a set into, in particular when the set is over a very large field e.g all integers The following question is one ...
Joe's user avatar
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How to prove that if two sets belong to the same sets, then they are identical?

The axiom of extensionality states that if two sets have the same members, then they are identical. In symbols, $(\forall x)(\forall y)((\forall z)(z \in x \leftrightarrow z \in y) \rightarrow x=y)$. ...
user107952's user avatar
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3 votes
1 answer
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Proof $[c]^{\omega}=\{A\subseteq c : |A|=\omega\}=c$

I would like to calculate the cardinality of the set $[c]^{\omega}=\{A\subseteq c : |A|=\omega\}$, where $c$ is the cardinality of the continuum (cardinality of the Reals) and $\omega$ is the ...
vinipenalty27's user avatar
2 votes
0 answers
40 views

Why $E$ shold be open in Baby Rudin theorem 9.19

I didn't understand why $E$ should be open or what part of the proof it is used Theorem 5.19 is If $f$ is a continuous function from $[a,b]$ to $R^k$ and $f$ is differentiable in $(a,b)$ then $\left|\...
Mathematics enjoyer's user avatar
3 votes
4 answers
89 views

Why do we want topologies to be closed under both finite and infinite union but not infinite intersection? [duplicate]

So I recently read the definition of a topological space and a topology from a book, and according to it, the topology must be closed under finite and infinite union, but it must only be closed under ...
zlaaemi's user avatar
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Show that $ |\mathcal{P}(\mathbb{N})|=\mathfrak{c}$

Show that $ |\mathcal{P}(\mathbb{N})|=\mathfrak{c}$ Attempt: In order to prove that $\lvert\mathcal{P}(\mathbb{N})\rvert=\mathfrak{c}=\lvert\mathbb{R}\rvert$, we know $\mathbb{R}\sim (0,1)\sim\mathbb{...
LJNG's user avatar
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Szekeres problem 1.5

In Peter Szekeres's text "A Course in Modern Mathematical Physics", problem 1.5 asks us to prove: $$ \begin{equation}\tag{1} A - \bigcup \mathcal{B} = \bigcup \{A - B_i \mid i \in I\}, \...
MattHusz's user avatar
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-2 votes
0 answers
24 views

Partition group in subsets not containing their inverses [closed]

Let $G$ be a locally compact group and $A=\{a\in G|a^{-1}\neq a\}.$ Is it always possible to partition the set $A$ in two disjoint measurable subsets $A_1,A_2$ such that $a^{-1}\not\in A_i$ for every $...
MathNewbie's user avatar
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1 vote
1 answer
33 views

How many pairs $(B, C)$ of such sets exist?

Suppose that $A = \{a, b, c\}$ and that we're choosing two subsets $B, C$ of $A$ so that $B\cap C = \emptyset$. How many pairs $(B, C)$ of such sets exist? If $|B| = |C| = 2$, then there must be a ...
Noether's user avatar
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0 answers
83 views

Is there a name for this set-theoretical definition of natural numbers, or has it been invented?

I'll call it the binary encoding with sets. I think it's nice and trivial, should have been discovered by many genius brains, but i can't find it by searching with efforts. Prior arts are Zermelo's ...
Farter Yang's user avatar
2 votes
1 answer
51 views

Solving a Set Theory Problem Involving Intersections, Unions, and Complements

I'm struggling with a particular set theory question and would greatly appreciate some guidance. The problem is as follows: Let $A=\{1,2,5,6,7\}$, $B=\{0,4,6,7,9\}$, and $C=\{0,1,2,6,7,9\}$ be subsets ...
Bishop_1's user avatar
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44 views

Showing a certain rule defines a topology

Let $L_1, L_2, L_3, \ldots$ be a family (sequence) of parallel lines in the plane, and let $X$ be an infinite union of $L_n$. Let us call a subset $G$ of $X$ big if either $G = \varnothing$ or $L_n \...
user1033615's user avatar
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0 answers
77 views

Interpreting statements in a proof which have implications for the non-emptiness of sets (sets, logic, cartesian products) - Tao Analysis I 3.5.6

This question is about deriving requirements for non-emptiness of sets from the algebraic steps in a proof. It is not primarily about the proving the primary objective of the exercise. Why am I asking ...
Penelope's user avatar
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Demonstrate that A is a countable set.

Question: Let $A = \{x \in \Bbb N \mid \exists y(x = 2y \lor x = y^2)\}$. Construct a surjective mapping $ f: \mathbb{N} \rightarrow A $. By doing this, you demonstrate that A is a countable set. ...
peterparker321's user avatar
1 vote
0 answers
36 views

Expressing the event $ \{ \limsup_{n \rightarrow \infty} X_n = l \}$

Let $(\Omega, \mathcal{H}, \mathbb{P})$ be a probability space. Let $(X_n)_{n=1}^{\infty}$ be a sequence of real valued random variables s.t. $\forall n \geq 1 \quad X_n$ is $\mathcal{H}$-measurable. ...
Fran712's user avatar
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0 answers
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Show that the interval ⟨2, 5⟩ ⊆ ℝ⁺ is an uncountable set.

Question: Show that the interval $⟨2, 5⟩ ⊆ ℝ⁺$ is an uncountable set.\ To show that the interval $ \langle 2, 5 \rangle \subseteq \mathbb{R}^+ $ is an uncountable set, we can use Cantor's diagonal ...
peterparker321's user avatar
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0 answers
55 views

Finite cardinals raised to the power of an infinite cardinal

I am trying to prove the fact that if $a$ and $b$ are finite cardinals, and $c$ is an infinite cardinal, then $a^c = b^c$. I am able to prove this fact by using $d \cdot d = d$ for all infinite ...
Mark Worrall's user avatar
0 votes
1 answer
104 views

Szekeres proof of set distributive law inadequate?

I'm reading "A Course in Modern Mathematical Physics" by Peter Szekeres. In problem 1.1, he asks the reader to show the distributive law $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$. The ...
MattHusz's user avatar
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0 votes
1 answer
84 views

proof of part of Tao Analysis I 3.5.6 too simple? (sets, cartesian products, logic)

I came up with a simpler solution to one part of Tao's Analysis I 4th ed exercise 3.5.6. Question: Several online solutions, and the one I originally developed (with help), use proof by contraposition,...
Penelope's user avatar
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0 votes
0 answers
22 views

How to describe the transitive closure of a relation in terms of the ground relation

Let $X$ be a non-empty set and $\equiv$ a relation on $X$, which is symmetric and reflexive but is not transitive. I know that there exists a transitive closure of $\equiv$, saying, $\equiv_{cl}$. But ...
Kaique Roberto's user avatar
-2 votes
1 answer
52 views

Determining whether the sets $\{a\in\mathbb Z\mid0\leq a\leq10\}$ and $\{b\in\mathbb N\mid b^2\leq100\}$ are equal [closed]

Let $$A = \{a \in \mathbb Z\mid 0 \leq a \leq 10\}$$ and $$B = \{b \in \mathbb N \mid b^2 \leq 100\}$$ Determine whether $A = B$. If they are equal, prove it. If they are not equal, provide a set $A^\...
Frances C's user avatar
1 vote
1 answer
91 views

Can't find step in proof that requires sets to be non-empty (sets, cartesian products, quantifier logic)

This question is about where in a proof I need to assert that a set is non-empty. The following is Exercise 3.5.6 from Tao's Analysis I 4th ed. Let $A$, $B$, $C$, $D$ be non-empty sets. Show that $A \...
Penelope's user avatar
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-1 votes
0 answers
29 views

Does the Axiom of Choice imply that for each element in a set, there is a choice function that would return it? [closed]

Let U be an uncountable set. Is the following true? $AoC \implies (\forall u \in \textbf{U}) \: \exists f: f(U) = u$ For example, given the set of Reals $\textbf{R}$, can we know that for every $r \in ...
Daniel Kondratov's user avatar
0 votes
1 answer
27 views

Is the dual of a principal filter to be a principal ideal?

A filter $F$ on $S$ is principal if $$ F=\{X\subseteq S\mid X\supseteq X_0\} $$ for some nonempty $X_0\subseteq S$. Consider $P(S)$ to be a Boolean ring $(P(S),\cup,\cap)$, then the $dual$ of a filter ...
BlowingWind's user avatar
1 vote
0 answers
35 views

Real number set in roster form. [closed]

I have read in a textbook that the set of real numbers , $\mathbb R$ , can not be represented in tabular or roster from. (Tabular or Roster representation is when elements of the sets are put in curly ...
Aryan Kr.'s user avatar
1 vote
0 answers
71 views

Is there any other way to make a $f :[0,1]\to \mathbb{R}$ discontinuous at every point of its domain?

The Dirichlet function: $$f(x) = \begin{cases} 0, & \text{if $x$ is rational} \\ 1, & \text{if $x$ is irrational } \end{cases}$$ is an example of a function that is discontinuous everywhere ...
pie's user avatar
  • 4,396
0 votes
0 answers
24 views

Relations such that each member of the range is the unique value of some member of the domain

A relation $R$ may have the property that each element $y$ of $\mathrm{ran}\, R$ is the unique right value of $R$ at some object, that is, for some object $x$, $\langle x,z\rangle \in R$ if and only ...
A. Burrell's user avatar
1 vote
1 answer
58 views

Which of these 4 statements about sets A and B are equivalent?

Which of these 4 statements about sets A and B are equivalent?: (1) A $\cap$ C $\subseteq$ B $\cap$ C for all C (2) A $\subseteq$ B (3) A $\cup$ C $\subseteq$ B $\cup$ C for all C (4) A \ B = $\...
Thao Mai's user avatar
1 vote
1 answer
74 views

Can we refer to "infinity" as a property of a set in ZFC/FOL?

I understand we have the Axiom of Infinity and have "access" to an infinite set, but is there a way to say in first-order logic "set A is infinite" or refer to its cardinality? The ...
lucas's user avatar
  • 39
0 votes
1 answer
94 views

Axiomatic set theory: problems understanding union of a set $\bigcup X$

From Jech p 9 (Set Theory, 3rd edition ) if $X$ is a set then $Y=\bigcup X$ is a set which is stated in formulas as; \begin{align*} \forall X\exists Y\forall u\in Y\leftrightarrow\exists z(z\in X\land ...
dandar's user avatar
  • 980
1 vote
1 answer
45 views

Can I just copy/paste the Axiom of Infinity formula without using it as an actual axiom to define an infinite set? (a developer asking)

Sorry for the weird wording of the title, happy to work on it. This is a continuation of Why is the Axiom of Infinity necessary? but from the perspective of a complete novice to axiomatic set theory (...
lucas's user avatar
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