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, ...

learn more… | top users | synonyms

1
vote
1answer
36 views

Elementary Set Theory: Verifying uniqueness of isomorphism between ordered sets

Let $W$ be a set and $E$ be a binary relation on $W$ satisfying the following properties: (a) For all $x, y \in W$, if for every $z \in W$, $z \in x$ if and only if $z \in y$, then $x = y$. (b) $E$ ...
0
votes
1answer
19 views

Why are both parts of $\forall a, b \in X,\ a R b \lor b R a.$ necessary for the definition of Total Order

On this wiki page it has this definition of a total order: $$\forall a, b \in X,\ a R b \lor b R a.$$ My understanding of $a$ and $b$ is that if $R$ is the relation of less than or equal to on ...
0
votes
0answers
27 views

Finding an equivalence relation given an undefined set

I need to define a relation $R$ on the set $X$ with a cardinality of $10$ with 10 elements such that $R$ is an equivalence relation on $X$. I can either define a property on $X$ or simply list the ...
0
votes
2answers
35 views

Problem to understand $f(A\backslash B)\neq f(A)\backslash f(B)$

I know that $f(A)\backslash f(B)\subset f(A\backslash B)$. Indeed, $$y\in f(A)\backslash f(B)\implies \Big(\exists x\in A\backslash B: y=f(x)\Big)\implies y\in f(A\backslash B).$$ But to me, $$y\in ...
3
votes
2answers
101 views

Full classes in Kelley's book

In Kelley's "General topology" (in the "Appendix") the full classes $X$ are defined as those with the property $$ \forall A\in X\quad A\subseteq X. $$ In the Russian translation it is added that this ...
3
votes
1answer
61 views

Elementary Set Theory: Ordinals, Well Orderings and Isomorphisms

I need to show that for any countable ordinal $\alpha$ there is a set A $\subseteq \mathbb{Q}$ such that (A, <) is isomorphic to ($\alpha, \in$). To do it I am supposed to show the following ...
2
votes
0answers
29 views

Associative set operation? [duplicate]

Define $A+B=(A\cup B) \backslash (A\cap B)$. Is this set operation associative? I've try to expand $(A+B)+C$ and $A+(B+C)$, but it just became too messy...
0
votes
1answer
28 views

The set of points in $\Omega$ which belong to exactly $k$ events is an event

This exercise if taken from Probability: An Introduction by Grimmett and Welsh. In what follows, $\Omega$ is a set and $\mathcal{F}$ is an event space of subsets of $\Omega$ (that is, $\mathcal{F}$ is ...
0
votes
1answer
27 views

Is this set Reflexive, Symmetric or Transitive?

Given the set: $$R = \{(x,y)\;|\;x\times y \leq 2015 \;\;\text{and}\;\; x,y \subset \Bbb N \setminus \{0\}\}$$ Would this set be just symmetric as I think? Or am I misunderstanding set relations?
-1
votes
3answers
45 views

Textbooks for Groups & Rings [closed]

Please I need suggestions on the best textbooks to help me comprehend this Groups and Rings and relate it with the rudimentary aspect of Set Theory
0
votes
3answers
32 views

Subset properties involving infinity

I am given that: $$n>3, n \in \Bbb N \;\;\text{and}\;\; A = \{1,2,3,...,n\}$$ And need to find how many subsets $B$ of $A$ have the property that: $$B \cap \{1,2\}=\emptyset$$ As well as how many ...
7
votes
1answer
110 views

cardinality of a basis for a topology

Suppose X is a space of cardinality $\le \kappa$. I would like to claim that any topology on X has a basis of cardinality $\le \kappa$. Intuitively it's true since even the discrete topology has such ...
0
votes
0answers
14 views

Functions to represent set operations?!

Assume you have set of real positive numbers $a_1,...,a_n$. And a strictly decreasing convex function $f$. Assume the intervals $A_i = [0,f(a_i)]$ to represent $i^{th}$ set, $i = 1,...,n$. Can we ...
1
vote
2answers
38 views

In a Set Definition for 2 elements, what does $\Bbb N \times \Bbb N$ mean?

I am asked how many elements are in the set: $$\{(a,b) \mid a,b\in\Bbb N \times \Bbb N \;\;\text{and}\;\; 1\le a\le b \le 15\}$$ And I assumed it would be the triangle number of 15 until I saw the ...
-1
votes
1answer
36 views

Does the set $[0.1)$ have the least upper bound property?

I know that any proper subset of $[0.1)$ has least upper bound "$a$" in $[0.1)$ If we consider $[0.1)$, which is improper subset of $[0.1)$, its least upper bound is $1$ which does not belong to set. ...
3
votes
0answers
16 views

Cartesian Product of a Union and an Intersection

I am given the Cartesian Product of an equation: $(A \cap C) \times (B \cup A)$ As being $\{(5,1),(5,4),(6,1),(6,4)\}$ And the sets: $B=\{1,9,4\}$ and $C=\{5,6,7,8\}$ And so I figure that $(A ...
2
votes
1answer
26 views

Probability function over infinite set

Hmm, I was just talking to a friend of mine...and I said that Personally I would like to define the discrete probability function to be $ |event|\over |sample space|$ Then I gave an example ...
0
votes
2answers
26 views

Set Subtraction with Self Reference

Is there an example of a set $A$ Where we let $B = \{1,3,4,8\}$ And $A - (B - A) = \{1,2,4,5,6,9,10,11\}$ I've been trying to get my head around it in the form of a Venn Diagram but it isn't going ...
1
vote
3answers
55 views

What is this set $\mathbb{R}$ mod $2\pi$

What does it mean for S = $\mathbb{R}$ mod $2\pi$? Can someone please explain as this notation is new to me.
0
votes
1answer
42 views

Are there Elements such that this Set Relationship is true?

I have a set containing a set denoted by: $\{\{?, ?\}\}$ And am looking to list elements, should there be any, such that: $ \{\{?, ?\}\} \subseteq \{1, 2, \{3, 4\}\} $ Any help would be ...
3
votes
1answer
73 views

Theorems not Formulable in Set Theory

Several sites I have been reading say that set theory is a good foundation for mathematics because virtually every theorem can be cast into a theorem in set theory. What is an example of a theorem ...
0
votes
2answers
23 views

Show $f^{-1}(A^c)=(f^{-1}(A))^c$ [duplicate]

Let $f: X \to Y$, and $A\subseteq Y$. Show that $f^{-1}(A^c)=(f^{-1}(A))^c$ I know how to prove that $f^{-1}(A^c)\subseteq(f^{-1}(A))^c$, but stuck on proving $(f^{-1}(A))^c\subseteq f^{-1}(A^c)$. ...
0
votes
1answer
24 views

Are the following families of sets closed under intersection?

Problem Statement Let $X$ be any set whatsoever, and let $f:X\to X$ be any function. Note that in general, no structure is imposed on $f$ whatsoever (i.e. continuity, linearity, etc). The problem is ...
2
votes
1answer
51 views

Cardinality of polynomials with real coefficients

What is the cardinality of the set of all polynomials with real coefficients? I know the power set of R is "more infinite" than R, so to speak, but I'm unsure of how to prove that there does or does ...
-4
votes
0answers
45 views

Set theory question [closed]

How to prove that $X \cap Y=Y$ if and only if $X+Y=X$? I said let $x \in X$ and $y \in Y$. Since $X+Y=X$, does it follow that $Y=\{0\}$?
0
votes
2answers
65 views

If $A\dot{-} B$ is countable and $B \dot{-} C$ is countable then $A\dot{-} C$ is countable? [closed]

Prove that: If $A\dot{-} B$ is countable and $B\dot{-} C$ is countable then $A \dot{-} C$ is countable? If not give a counter-argument
0
votes
1answer
17 views

Can you give me an example of x($X/\mathscr T$)y?

Definition 7. Let $\mathscr T$ be a partition of a nonemptyset X. We define a relation $X/\mathscr T$ on X by x($X/\mathscr T$)y if and only if there exists a set $A \in \mathscr T$ such that $x, y ...
0
votes
1answer
31 views

If $F$ is a one-to-one function, then if $y \in ranF$, then$ f(f^{-1}(y)) = y$

This is not the proof given in the lecture, but I found a way that seemed far more intuitive to me, so I wanted to check if it was right. $F$ is one-to-one, so there's only one $y : f(x) = y$. If $y ...
0
votes
1answer
24 views

Why $y/\mathscr E$ is an element of $X/\mathscr E$ when it's defined $X/\mathscr E=\{\,x/\mathscr E\mid x\in X\,\}$?

"Theorem 4 Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. Then $X/\mathscr E$ is a partition of $X$. [Proof] By Theorem 3(a) and Definition 6, $X/\mathscr E =\{\,x/\mathscr ...
0
votes
2answers
38 views

Show that $(A \cup C) \cap (B \cup C') \subset A \cup B$

I am trying to prove that $$(A \cup C) \cap (B \cup C') \subset A \cup B$$ So far I got: $$(A \cup C) \cap (B \cup C') \subseteq (A \cup C \cup B) \cap (B \cup C' \cup A)$$ $$(A \cup B \cup C) \cap ...
0
votes
0answers
24 views

Set, n-Tuple, Vector and Matrix — links and differences

I know this question has been asked like 1000 times, however all supplied answers were not really satisfying to me. My question concerns the similarities and differences between these mathematical ...
0
votes
1answer
25 views

What's the difference between $x/\mathscr E$={$y \in X$∣y$\mathscr E$x} and X/$\mathscr E$={x/$\mathscr E$∣$x\in X$}?

For example Let X={0, 1, 2, 3, 4} and $\mathscr E$ is an equivalence relation on X. $\mathscr E$ is defined as $\mathscr E$ ={(0, 0), (0, 4), (1, 1), (1, 3), (2, 2), (4, 0), (3, 3), (3, 1), (4, 4)}. ...
1
vote
3answers
71 views

Are “formulas” in Axioms of ZFC indefinite?

There is the Separation Schema in Axioms of ZFC. Where do "formulas" in this axiom comes from?Are they indefinite and is ZFC actually something like "ZFC(X)" where X is a variable which denotes a ...
1
vote
2answers
45 views

Set Theory Question: The proof of finite union to infinite union [closed]

I am sucked in the proof of if Union of A from n=1 to k is equal to Union of B from n=1 to k, then Union of A from n=1 to inf is equal to Union of B from n=1 to inf. I can intuitively think this is ...
1
vote
3answers
51 views

Dumb question, $A \subset B$, what is $A - B$

I know this is a super dumb question, but if you had a little set $A$ contained in a bigger set $B$, what the set difference $A - B$? $B - A$ is well defined and I can visualize it in my head ( donut ...
0
votes
3answers
41 views

Cardinality and order relations on $ \Bbb{C} $ and $ \Bbb{R} $.

It has been demonstrated that complex numbers have cardinality $ \aleph_{1} $. However, it can also be shown that the complex numbers cannot be made an ordered field. How can these two facts coincide? ...
-3
votes
2answers
126 views

Can a diagonal be longer than the list being diagonalized?

If we have a list of rational numbers like this... $\ 0.n_1,_1 $ where $\ n_1,_1 $ is a digit from $\ 0$ to $\ 9$ $\ 0.n_1,_2 n_2,_2 $ $\ 0.n_1,_3 n_2,_3 n_3,_3 $ $\ 0.n_1,_4 n_2,_4 n_3,_4 ...
2
votes
1answer
30 views

What is the difference between disjoint union and union?

If $S = A \cup B$, then $S$ is the collection of all points in $A$ and $B$ What about $S = A \sqcup B$?, I think disjoint union is the same as union, only $A, B$ are disjoint. So the notation is a ...
3
votes
1answer
27 views

Show that $X$ can be represented as a union of disjoint equivalence classes

Let $X$ be a set. Let $\sim$ be an equivalence relation on $X$. Show that $X$ is the union of disjoint equivalence classes $\{x\}$ for $x \in X$. What I have tried: claim: $X = \bigcup_{x\in X} \{ x ...
1
vote
0answers
41 views

Ideas for approaching set theory when you've already studied higher abstractions?

I've come to acknowledge (or so I think). That many of the concepts e.g. in real analysis (like, say, continuity), actually don't (in modern times) boil down just real analysis. But rather, e.g. set ...
0
votes
1answer
29 views

Set theory; sets and subsets; Is an empty set contained within a set that contains real numbers? [duplicate]

Here is the context for my question: Let A = {1,2,5,8,11}. Here is my question: Is ∅ ⊆ A? Why or why not?
0
votes
1answer
35 views

Evaluate the image of a function

I am given a function: $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}, \space\space f(x,y)=(x-y^2)(y-x^2)$. I have to evaluate the image $f(A)$ of a set $A=\mathbb{R}^+\times \mathbb{R}^+$. I tried ...
0
votes
1answer
34 views

What does it mean for the empty set to be connected and totally disconnected?

I am trying to prove that the empty set is disconnected, but every single post I can find on this topic is about showing empty set is connected. Recall definition of connected. A set $S$ is connected ...
2
votes
7answers
53 views

Show that $A \subset B \implies A \cap B = A$ [duplicate]

I am trying to show that: $$A \subset B \implies A \cap B = A$$ So far I got: $$A \subset B$$ $$A \cap B \subset A$$ $$A \cap B \subset B$$ $$A \cap B \subset A \subset B$$
2
votes
4answers
44 views

How to show that $A \cup B = B \implies A \subset B$

I am trying to show that $$A \cup B = B \implies A \subset B$$ but I get stuck on: $$x \in A \cup B = B$$ $$x \in B$$
0
votes
2answers
23 views

Prove that a denumerable set can be partitioned into two denumerable subsets

I was wondering if this "proof" is sufficient in demonstrating a that a denumerable set $A$ can be partitioned into two denumerable subsets $A_1$ and $A_2$. Let $A$ be a denumerable set and define $A ...
0
votes
2answers
22 views

Proving distributivity of Complement over union (set theory)

I am trying to prove the following identity: $$M \setminus (N \cup L) = (M \setminus N) \cap (M \setminus L)$$ I thought about saying that $x \in (N \cup L)$ which means that $x$ is in either $N$ or ...
0
votes
2answers
24 views

If a set $S$ has a proper subset $A$ that is infinite, then $S$ is infinite.

I am working through Herstein's Algebra and I have become stuck on a seemingly simple exercise. I am using the definition that a set S is said to be infinite if there exists a bijection between $S$ ...
1
vote
1answer
24 views

Calculate the number of equivalence relations $S$ that satisfies $R \subseteq S$

Let $A=\{1,2,3,4,5,6,7,8\}$ and let $R=\{(1,2),(5,4),(4,5),(6,2),(4,4),(6,5),(7,8)\}$ be a relation on A. What it the number of equivalence relations $S$ that satisfies $R \subseteq S$ I know what ...
1
vote
3answers
111 views

What do these symbols mean: $\bigcap$, $\bigcup$, $\bigwedge$, $\bigvee$?

I know that some of these symbols are used in set theory like $A \cup B$, but that's not what I'm talking about. I have seen those symbols used in a way similar to $\Sigma$ summation and $\Pi$ ...