Questions tagged [elementary-set-theory]

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, differences and complements, De Morgan's laws, Venn diagrams, relations and functions, countability and uncountability, etc. More advanced topics should use (set-theory) instead.

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Parenthesis needed? in $(S\subset T)\vee(S=T)\iff S\subseteq T$ [closed]

In the statement $$(S\subset T)\vee(S=T)\iff S\subseteq T$$ are the parenthesis formally incorrect correct but deemed best removed correct and optional correct and deemed required for readability ...
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Is $\{1,2,3\}$ equal to $\{\{1,2\},3\}$ since $\{1,2\}$ is a subset of $\{1,2,3\}$?

I have a problem with subsets in Set Theory at its most basic level. a set $A$ is a subset of set B if all the elements of $A$ are also elements of $B$. $\{1,2\}$ is a subset of $\{1,2,3\}$ Simple ...
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Power set $\mathcal{P}(A)$ with the $\subseteq$ relation is a partial ordered set, however it may not be a total ordered set

I have a question that i cannot understand. Question Let $A$ be a set. Show that the power set $\mathcal{P}(A)$ with the $\subseteq$ relation is a partial ordered set, however it may not be a total ...
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If the product topology on $\mathbb{R}^2$ is all sets of the form $(a,b)\times(c,d)\subseteq\mathbb{R}^2$ what is product topology on $\mathbb{R}^4$?

Probably the answer to this question is too obvious for it to be asked but I want to make sure my assumption is correct. Consider the real line $\mathbb{R}$ with the topology that has basis consisting ...
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$A,B$ such that $A\cap B=\emptyset$ and $A\cup B=\mathbb{R}$ and $B=\{x+y : x,y\in A\}$?

If set $A,B$ satisfy $A\cap B=\emptyset,A\cup B=I$, and $B=\{x+y : x,y\in A\}$, can $I$ be real number set $\mathbb{R}$? I think the answer is yes, but I can't construct it. If $A$ is odd number set, ...
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Prove for the ideals $I,J$ that $J\nsubseteq I$

consider the ideals $I=\langle f_1,f_2\rangle$ and $J=\langle h_1,h_2\rangle$ in $\mathbb{Q}[x,y]$. I want to prove that $J\nsubseteq I$. I'm trying to do this by proving that every element from $J$ ...
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1 vote
1 answer
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How to show that $\forall x \in \mathbb{R}\ \exists! n \in \mathbb{Z}$ such that $x \leq n < x+1$?

To show that: $$ \forall x \in \mathbb{R}\ \exists! n \in \mathbb{Z} \text{ s.t. }x\leq n < x+1 $$ I know that the said $n$ is the infimum of the set of integers greater than or equal to $x$: $$ n =...
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Shorthand set builder notation?

I was reading some papers and they had the following notation: $$ \big\{k\big\}_{k=0}^n $$ I assume this implies that $$ \big\{k\big\}_{k=0}^n = \{k \in \mathbb{WHAT} : 0 \leq k \leq n\} $$ What is $k$...
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For $f : \mathbb{R} \to \mathbb{R}$, what is $\sup\limits_{x \in A} f(x)$ if $A=\emptyset$? [duplicate]

Let $f : \mathbb{R} \to \mathbb{R}$ and $A$ be an empty set. What is $\sup\limits_{x \in A} f(x)$ equal to? I reckon it should just be undefined or is there an alternative convention?
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Finding a subset of the powerset whose union is a subset

Let $P_3(S)$ denote the powerset of $S$ where all elements of length greater than $3$ are removed. Given a subset of $S$, $E$; how many subsets of $P_3(S)$ $R$ are there such that the union of the ...
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2 answers
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Notation: function returning the element of a partition containing $x$

Suppose $Y=\{y_1,\ldots,y_m\}$ partitions the set $X=\{x_1,\ldots,x_n\}$. I would like to define a function $y: X \to Y$ which returns $y \in Y$ if and only if $x \in y$. Is there a way to write this ...
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2 votes
1 answer
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Show that the definition of Scott topology is a topology

To verify the definition of a Scott topology is a topology, I still need to show that it's closed under intersection. Can someone help? Definition 1 (Scott topology). Let $(D,\leq)$ be a complete ...
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1 answer
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Prove that a "set of all sets" does not exist.

Axiom I used for the proof: The Axiom Schema of Comprehension: Let P$(x)$ be a property of $x$. For any set $A$, there is a set $B$ such that $x\in B$ if and only if $x\in A$ and P$(x)$. Here is my ...
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Hi... I just beginning with the study of categories so my question might seem elementary. [closed]

Please in want to know how to show that the functor $\mathcal{P}(\Sigma \times Id) $ weakly preserve pullbacks.
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All subsets of well-ordered set are well-ordered - quibble or mistake?

Context: cheating on my homework like everybody else. Let $S$ be a well-ordered set under a relation $R$. Now there are books out there which say something like, for example: Clearly if $\le$ is a ...
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2 votes
1 answer
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Are there more real numbers in the interval $[1,\infty)$ than in the interval $(0,1]$? Or not?

We all might be familiar with the beautiful method Cantor devised to prove that the cardinality of the set of real numbers is more than that of the set of natural numbers (Refer to: https://en.m....
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consider the following relation on the set of positive integers.

consider the following relation on the set of positive integers. $R =\{ (x,y) |x+y >10 \}$ Is this relation reflexive? Justify your answer. Is this relation symmetric? Justify your answer. Is this ...
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2 votes
0 answers
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Help showing that two sets are equal.

Let $A,B\in\mathbb{N}$ and define $$\tag{1} U(A, B):=\{K \subseteq \mathbb{N} \mid A \subseteq K \subseteq \mathbb{N} \backslash B\} \subseteq 2^{\mathbb{N}} $$ Define further $f: 2^{\mathbb{N}} \...
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Cardinal of a set challenge

Denote by $[n]$ the set $\{1,\dotsc,n\}$, for any real $n$, and let $M_n$ be a positive increasing sequence, $M_n<n$. Consider the set $$S=\{(l_1,l_2,l_3,l_4)\in[n]^4:\forall a\in[4]:\exists b\in[4]...
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-1 votes
0 answers
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can we replace two elements if they are Isomorphic to each other.

If lets say I have two sets $A =\{a_{1},a_{2},a_{3}\}$ , $B =\{b_{1},b_{2},b_{3}\}$ if the two sets A and B are Isomorphic, then can I write $a_{1}= b_{1},a_{2}=b_{2},a_{3}=b_{3} $ or can i replace A ...
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-1 votes
1 answer
77 views

Is it meaningful to invert large cardinals?

As cardinal numbers involve the notion of ever increasing multitudes of things, is there a mathematically useful concept of ever decreasing multitudes? We already have rationals tending to zero, and ...
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1 vote
2 answers
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proof performed in inclusion of sets

I want to prove the following identity: $$(A\cup B)\cap (A\cup C)\subseteq A\cup (B\cap C)$$ From this equivalence I know that for a full test I should be able to prove that the left term is included ...
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Proving that the quotient of a set by an equivalent relation is a partition

I need to show that the quotient of a set $S$ with respect to the equivalence relation $\sim$ is a partition of $S$. To show this, we will denote the quotient by $P_\sim.$ Note that $$ P_\sim = \{[a]_\...
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-2 votes
2 answers
33 views

Notation: set X whose elements are smaller than each elements of set Y

Does there exist some notation to indicate that all elements of a set X are smaller than all elements of a set Y?
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2 votes
1 answer
57 views

Prove that a relation R on set A is antisymmetric if and only if $R \cap R^{-1} \subseteq \{(a,a):a \in A\}$.

Can someone check to see if my proof is correct? If it actually is correct, can someone tell me how to be less verbose and "make it mathy and less wordy" for my backwards implication $(\...
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-2 votes
0 answers
52 views

Is $\{ y \mid y\in f(x), x \in L\}$ equal to $\bigcup\limits_{x\in L}f(x)$?

I have one set $S_L = \{ y \mid y\in f(x), x \in L\}$ and another set $S'_L = \bigcup\limits_{x\in L}f(x)$. Are these two sets equal?
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0 votes
1 answer
18 views

Notation for adding elements to ordered set

Say I have an ordered set $E_1 = (e_1, e_2, ..., e_j)$ composed of some elements. I'd like to add a few more elements $E_2 = (e_k, e_{k+1}, ..., e_\ell)$ to this ordered set so as to obtain set $E_3 = ...
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0 answers
79 views

Does a set of all decimal expansions of $\pi$ contains $\pi?$ [duplicate]

Let's say there is a set containing all finite decimal expansions of $\pi$: $$A = \{3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... \}$$ Does this set contains $\pi$? I see that it is probably not true ...
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Trouble understanding a simple exercise about primitive recursion on natural number set theory.

Consider the function $f: \omega \rightarrow P\omega$ defined recursively through: \begin{equation*} \omega \in P\omega \quad \text{and} \quad h : \omega \times P\omega, (n,A) \rightarrow \{n^+ + k \...
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0 votes
2 answers
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What does it mean to say "the 'less than' relation on the set D is the set{(a, b): a, b $\in$D and a<b}"?

I'm studying "Introduction to Algorithms, 3rd Edition by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein." Section B.2 of the book gives an example of binary ...
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1 vote
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Prove that if a is an infinite cardinal, then for each finite cardinal n, $a^n = a$

The proof in my textbook is the following: $a^n = card\:A^n$, the set of all maps of $I_n$ into a set $A$ of cardinal $a$. Since $A^{n} \subset I_n \times A$, $a^n \leq na = a$. On the other hand, for ...
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0 votes
1 answer
36 views

How to systematically list products of sets

Problem: Suppose I have three sets $A$, $B$, and $C$. Consider the product $\{A, A^c\} \times \{B, B^c\} \times \{C, C^c\}$. There are eight elements in this product and I want to list them ...
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1 vote
1 answer
38 views

I need some assistance in certain sections of Naive set theory by Paul R Halmos [closed]

So I (16M) was studying the aforementioned books and couldn't grasp certain parts of it . Could someone please explain these parts more explicitly. here goes- I understood the Axiom of Extension in ...
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1 vote
1 answer
54 views

Prove that $\{x\in A:P(x)\}$ is always a subset of $A$

I'm trying to prove the following, Proposition. Let $A$ be a set and let $P(x)$ be a property pertaining to $x$, then $\{x\in A:P(x)\}\subseteq A$. Here is my proof: proof. To prove $\{x\in A:P(x)\}\...
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0 votes
1 answer
34 views

Doubt on how to represent the Complement of a set in relation to another [closed]

I'm not sure how to represent a set using a property of the element, well I'm thinking of $3$ ways to represent this, the first way seems to be wrong, because I'm not using the variable $x$ to ...
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1 vote
2 answers
82 views

What are some examples of "Jordan spaces" which are *not* homeomorphic to a subspace of $\Bbb R^n$ (with the Euclidean topology)?

Note: This question has been substantially revised, see the edit history for earlier versions. So far, the only examples of metric spaces which I have seen in topology books are Euclidean $n$-space, ...
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Bijection between the set of strictly increasing sequences of natural numbers and the set of all sequences of natural numbers [closed]

We have two sets: $A = \{f \in \mathbb{N}^\mathbb{N}:\ f \text{ is strictly increasing}\}$, $B=\mathbb{N}^\mathbb{N}$, i.e the set if all sequences of natural numbers. I'm looking for a bijection ...
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-1 votes
0 answers
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Set Theory - True Or False [closed]

i have this question, which I need to decide if this expression is T or F for every given X,Y sets P(X ∩ Y) ∩ P(X ΔY) = {Ø} I can only assume it’s T but I have a real issue with proving something to ...
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1 vote
1 answer
17 views

An identity about Probability of unions

Let $(\Omega,\mathfrak{A},P)$ be a probability space and events $A_1,...,A_n \in\mathfrak{A}$ with $P(A_{i_1} \cap...\cap A_{i_k})=P(A_1 \cap...\cap A_k)$ for all $k \in \{1,...,n \},i_k\in\{1,...,n \}...
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0 votes
0 answers
31 views

Basic properties of equality for sets

I'm trying to prove the following proposition, Proposition [Basic properties of equality for sets] Let $A,B,C$ be sets. Thus the following holds, (Equality is reflexive) $A=A$. (Equality is symmetric) ...
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1 vote
1 answer
52 views

Reference for Analysis book in which natural numbers constructed from sets

Could anyone suggest books on Mathematical/Real Analysis that construct natural numbers through sets not Peano axioms? I find construction of natural numbers through sets more convenient. So I ...
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0 answers
26 views

Describing the set of elements that are in at least one of the sets A or B

Textbook problem: Use unions, intersections, and complements to express the set of elements that are in at least one of the sets $A$ or $B$, where $A$ and $B$ are subsets of the set $\Omega$. My ...
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1 vote
2 answers
114 views

How to prove $2n+1$ is odd for $n \in N$?

Usually, it seems that if $x\in N$, then "$x$ is odd" is translated by definition as $\exists y\in N: x=2 y+1$, but can we prove this? Given: $\forall a\in N:[Even(a)\iff \exists b\in N: a=...
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2 votes
1 answer
219 views

Proof that $\inf(A)=-\sup(-A)$

Prepping for a master's program in pure mathematics. I'm working on my problem-solving skills and was hoping someone would kindly verify my proof. Let $A$ be a nonempty set of real numbers which is ...
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1 vote
1 answer
31 views

How to express formally all possible 2-tuples of a set into one set

Let be a triplet of sets $(A, B, R)$, where: $A = \{a, b, c, ...\}$ $B = \{1, 2, 3, ...\}$ $R$ denoted the set for relations for the elements of the sets $A$ and $B$, $R = \{(a, 1), (b, 2), (c, 3), .....
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0 votes
1 answer
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Name and notation of the sum of cardinalities of finite sets

Given a finite set $S = \{s_1, s_2, ..., s_n\}$, where each $s_i$ is a finite set of $|s_i|$ elements, let $$ N = \sum_{s \in S}{|s|} $$ Is there a canonical name and notation for $N$? On a ...
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1 vote
1 answer
25 views

Composition of a relation with its inverse

I'm self-learning my way through Set theory and came across this question. Now I have a few difficulties to gain an entry to this question. The textbook and the lecture videos which I'm using gives ...
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2 answers
35 views

Proof that $f^{-1}(\overline{A}∩B)⊆f^{-1}(\overline{A})∩f^{-1}(B)$

Prove $f^{-1}(\overline{A}∩B)⊆f^{-1}(\overline{A})∩f^{-1}(B)$ Where f: P→Q and A and B are non empty subsets of Q So far I have: $f^{-1}(\overline{A}∩B)⊆f^{-1}(\overline{A})∩f^{-1}(B)$ $f^{-1}(\...
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-1 votes
1 answer
43 views

Dedekind Cut and Real Numbers

In defining real numbers with a Dedekind Cut there is an issue that continues to confuse me. Suppose we perform a Dedekind cut on the number line, and we look at say the left set of rationals. (Where ...
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1 vote
1 answer
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Let $\mathcal{A}= \{A_n = \left(0, \frac{n}{n+1} \right) \ | \ n \in \mathbb{N} \}$. Prove that $\bigcup \mathcal{A}\subseteq (0,1)$.

Let $\mathcal{A}= \{A_n = \left(0, \frac{n}{n+1} \right) \ | \ n \in \mathbb{N} \}$. Prove that $\bigcup \mathcal{A}\subseteq (0,1)$. Can you verify this solution? Let $x \in \bigcup \mathcal{A}$. ...
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