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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3
votes
2answers
35 views

Notation for union / intersection (in the same way $\pm$ stands for plus / minus) - is this a good idea?

Note: $F$ is a class of sets. I was solving a problem in Apostol's Calculus Volume 1. It is to show that $$B-\bigcup_{A\in F} A=\bigcap_{A\in F}(B-A)\qquad\text{ and }\qquad B-\bigcap_{A\in F} ...
2
votes
2answers
46 views

What is $A-B\cup C$ in words?

I'm working through the set theory exercises in Apostol's Calculus Volume 1 and am having some trouble describing $A-(B\cup C)$ in words. What I'm thinking is: If $x\in A-(B\cup C)$ then $x$ is in $A$ ...
0
votes
3answers
40 views

Bogus set theory proof

I'm having trouble figuring out where I went wrong in this proof. I think it's to do with my understanding of things like $\cup$ and $\cap$ in that I don't really have a solid understanding of what ...
1
vote
2answers
65 views

Prove that the Cardinality of $| \Bbb R \times \Bbb Z |$ has the same cardinality of $\Bbb R$

Prove that $| \Bbb R \times \Bbb Z | = |\Bbb R|$ So I know I have to prove that there exists a bijective function between $\Bbb R \times \Bbb Z$ and $\Bbb R$. How I would do that, I don't know. I ...
4
votes
1answer
65 views

Set-theoretic equality

Let $A⊂U^{*},B⊂U.$ Find the set $X⊂U,$ that satisfies the equation. $$(\overline{X \cup A}) \cup (X \cup \overline{A}) =B.$$ My thoughts: $$\begin{align}B&=(\overline{X \cup A}) \cup (X ...
2
votes
2answers
155 views

Definition for the set of Real Numbers

Could the set of Real Numbers be defined as \begin{array}{l} \mathbb{R} \equiv \mathbb{Q} \cup \{ x\neq \frac{a}{b} :a\wedge b\in \mathbb{Q} \} \end{array} ? Why or why not?
0
votes
3answers
34 views

Why ordered sequences can be reduced to sets?

I am trying to understand why ordered sequences can be reduced to basic sets. I understand most of the following proof: Sequences can be defined as functions Functions are a special case of ...
4
votes
3answers
41 views

Discrete math - Set theory - Symmetric difference: Proof for a given number.

I can't find anything on this topic elsewhere. I'd like to know what keywords/sites I should be using to find what I'm looking for if this is to elementry of a question. (been using discrete math, set ...
0
votes
4answers
2k views

Venn Diagrams Set Theory Explanations

I was hoping someone could explain how these sets map out on the venn diagram. Mainly, i am confused on what the Triangle means in terms of the venn diagram. (The Triangle represents: The symmetric ...
1
vote
5answers
49 views

Proving that that ${(R \setminus S)\setminus T} \subseteq R \setminus (S \setminus T)$

How might I prove that ${(R \setminus S)\setminus T} \subseteq R \setminus (S \setminus T)$? I am not sure the best place to start other than assuming $x\in(R \setminus S)\setminus T$ and trying to ...
1
vote
1answer
31 views

why this is not transitive yet a reflexive relation?

The relation given is $ R = \{(a,b); 1ab>0; a,b ∈ R \} $ I clearly understand that this is symmetric since $a*b = b*a$ but I'm not able to understand that why is this reflexive also and not at all ...
0
votes
1answer
14 views

Preference relations and the existence of extensions of functions representing them

In a book I found the following question: Let $\succsim$ be a complete preference relation on a nonempty set $X$, and let $\varnothing \neq B \subseteq A \subseteq X$. If $u \in [0,1]^A$ ...
0
votes
0answers
13 views

Indexed sum of cardinals [duplicate]

Let $\{ \kappa_i | i\in I \}$ be an indexed set of cardinals. We define the sum as: $\sum\limits_{i\in I}\kappa_i = \left\vert \bigcup\limits_{i\in I}X_i \right\vert$, where $\left\vert X_i ...
2
votes
2answers
66 views

Prove that $E \cap E^{c} = \varnothing$.

This is a 'simple' question on elementary set theory. I said 'simple' because statement like this is presented all over introductory sections in advanced math books, but they are not really proved in ...
0
votes
1answer
14 views

Draw figures for the 5 different lattices with 5 elements.

I can only think of 4 lattices. Those are: a 1-1-1-1-1 (a chain), a 1-3-1, a 1-1-2-1 and a 1-2-1-1 (if this notation isn't clear, I'll provide images). I really can't figure out what the 5th lattice ...
3
votes
1answer
29 views

The set $T=\{l\in\mathbb{N}: ml=nl \ \text{implies} \ m=n \}$ is inductive.

I'm trying to prove the following statement: $ml=nl$ implies $m=n$ for every $m,n,l\in \mathbb{N}$. So I defined the set $T=\{l\in\mathbb{N}: ml=nl \ \text{implies} \ m=n \}$ and if I prove that ...
0
votes
1answer
25 views

venn diagram and overlapping set equation

Q) Of the 24 dogs attending puppy school -six are small -twelve are brown -fifteen have long hair -one is small and brown and has long hair -two are small and brown but their hair is not long ...
1
vote
2answers
35 views

Predicate logic inference in a simple proof of uniform continuity.

For a function $f$ from a metric space $X$ into a metric space $Y$, uniform continuity can defined in this way: $\forall ε>0:\existsδ > 0:\forall p,q\in X:d_{X}(p,q)<δ \rightarrow ...
3
votes
3answers
41 views

Prove that equality holds only if $f$ is one-to-one.

I am just looking for a hint. Not a solution as I am just trying to solve these for fun. Let $f:A \rightarrow B$ with $A_0 \subset A$ and $B_0 \subset B$. Show that $$A_0 \subset f^{-1}(f(A_0))$$ ...
1
vote
3answers
53 views

Union of sets proof

Prove that $\{3t\}\cup\{3t+1\}\cup\{3t+2\}=\Bbb Z$, where $t$ is in the set of integers. It makes sense that you can get every integer from this Union of sets but how would you prove something like ...
2
votes
1answer
52 views

Easiest way to find the 'area of a Venn diagram,' given certain information.

We have a bunch of intersecting regions: $$X_1,\dots, X_n,$$ all with non-negative volume, and we know $V(X_i)$ and $V\left((\cup_{a\in A}X_a)\cap (\cup_{b\in B}X_b)\right)$ for any disjoint ...
-6
votes
0answers
22 views

Bijection of finite and infinite sets [on hold]

Prove that a nonempty set T1 is finite if and only if there is a bijection from T1 onto a finite set T2.
1
vote
3answers
70 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
2
votes
1answer
45 views

How to Prove It Exercise 7.2.5

Prove that ${}^{\mathbb{Z}^+} \mathcal{P}(\mathbb{Z}^+) \sim \mathcal{P}(\mathbb{Z}^+)$ where ${}^A B$ means the set of all functions $f:A \rightarrow B$ and $\mathcal{P}(A)$ is the power set of $A$. ...
3
votes
2answers
130 views

The set of all real functions of a real variable

How can I prove that the set of all real functions of a real variable, or even that the set of functions that take only the values 0 and 1, more than the continuum? I have one idea, but ...
1
vote
2answers
41 views

counterexample in relations of sets

Suppose $R$ is a relation from $A$ to $B$ and $S$ and $T$ are relations from $B$ to $C$. Can anyone produce a counterexample to $(S \setminus T)◦R⊆(S◦R) \setminus (T◦R)$?
-1
votes
1answer
28 views

Determining sets using basic operations

Let A={ O, {O}, 1, a, cat, {1, a, cat}} where O has been used to represent the null set. Determine the follwing: (a) A \ {a, b, c} = {O,{O}, 1, cat, {1, a, cat}} (b) AU{X}= {O,{O}, 1, a, cat, {1, a, ...
0
votes
2answers
40 views

Cardinality of a set containing sets

I've just started learning basic set theory and am puzzled by this question I came up with: What is the cardinality of {1, {2,3}}? Do I treat sets within sets as just one element and so the answer ...
4
votes
1answer
49 views

Cardinality of a set of natural sequences

Let $a=(a_n)_{n\ge 1}$ a sequence such that for every $n\ge 1$ we have: a) $a_n \in\mathbb{N}$ b) $a_n\lt a_{n+1}$ c) Exists $\displaystyle\lim_{n\to \infty} \frac{\#\{j\mid a_j\le n\}}{n}$ Let ...
3
votes
4answers
97 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...
1
vote
1answer
42 views

Set-builder Notation

In set-builder notation we describe a set in the following way: $A=\left\{x:\phi (x)\right\}$ Is it correct to say the following? Fix any $x_{0}\in X$ Evaluate the predicate $\phi(x_{0})$ ...
0
votes
2answers
31 views

Totally ordered $\sigma$-algebras

I know that every $\sigma$-algebra is partially ordered with respect to the inclusion operator $\subset$. However, it seems as though every $\sigma$-algebra should be totally ordered with respect to ...
1
vote
2answers
34 views

Triangular number method - Hilbert's hotel

There is a hotel with and infinite number of numbered rooms, each occupied by a single guest. An train with an infinite number of (numbered) coaches, each with an infinite number of (numbered) seats, ...
1
vote
2answers
252 views

Question about intersection/union of a set and its complement

I was answering this multiple choice question from this website examtimequiz.com/maths-mcq-on-sets: If $A$ is any set, then $A \cup A' = U$ None of these $A \cap A' = U$ $A \cup A' = ...
1
vote
1answer
38 views

Intersection of Countably Infinite Sequence of Sets [on hold]

Suppose $\{\Omega_k\}_{k=1}^{\infty}$ is a sequence of sets, where $\Omega_k$ is countably infinite and $\Omega_{k+1}\subset\Omega_k$ for all $k$. Is it possible to show that $\cap _{k=1}^{\infty} ...
-4
votes
0answers
30 views
1
vote
0answers
24 views

How to prove partial ordering formally?

The question is: The set $S$ is defined as $\varnothing \in S$, If $x \in S$, then also $\{x\} \cup x \in S$. Prove or disprove it is partial ordering. So the set $S$ looks ...
4
votes
1answer
33 views

Confused about a well-ordering lemma

I happened to stumble across the following lemma in Kenneth Kunen's set theory book: $\textbf{Lemma:}$ Let $\langle$ $A,R$ $\rangle$ be a well ordering. Then for all $x \in A$, $\langle$ $A,R$ ...
0
votes
1answer
38 views

Injective function, $f:X\to X$ with $f(X)\subset X$, but $T\subseteq X$ is not inductive set.

I'm looking for an example of the following manner: Suppose that $f:X\to X$ is a injective function(where $X$ some set), such that the following property not holds: If $T$ is subset of $X$, with ...
-1
votes
1answer
51 views
0
votes
1answer
383 views

A problem on Venn diagram

Please does the following problem requires three overlapping circles or just two to display the information in a Venn diagram? I need some hints. A stock broker was presented with the detailed ...
1
vote
2answers
14 views

A finite set and the set of its fixed points under any involution have cardinalities of the same parity

I am trying to write down a formal proof of the following fact: Let $A$ be a non-empty finite set and $f$ an involution on $A$. If $A'$ is the set of fixed points of the involution $f$, then $|A| ...
-1
votes
1answer
14 views

How to prove: Union of a chain of well-ordered sets w.r.t continuation is well ordered [on hold]

How to prove: Union of a chain of well-ordered sets w.r.t continuation is well ordered
0
votes
0answers
22 views

With a sequence $\{B_n\}$ and a function defined on all of its elements, what are the spaces between the outputs of the function?

I have a sequence $\{B_n\}$ and a function defined for every member of that sequence: $f(B_i,C_j)=a_j^i$ (Where the spaces between any two adjacent $C$'s is always constant). Such that the following ...
1
vote
1answer
43 views

Logical equivalence - Russell's Paradox

In 'How to Prove it' Velleman creates the following set: $R = \{A\in U| A \notin A \}$. This is, according to Velleman, equivalent to $\forall A \in U (A \notin A \iff A\in R) $. That is clear. ...
0
votes
2answers
60 views

Confusion about the definition of reflexive relation

The definition of a reflexive relation over $A$ is: $R$ is reflexive over $A$ iff $\forall a \in A :(a,a) \in R$ Why the '$\forall a \in A$'? Def. of transitive and symmetric relations don't have ...
3
votes
1answer
27 views

The set is closed (resp. open) iff the complement set is open (resp. closed)

There's a theorem in my small danish course book. Let $(M,d)$ be a metric space. Theorem: The concepts of open and closed are dual: A set $A\subseteq M$ is closed (resp. open) if and only if the ...
5
votes
1answer
2k views

Prove that $\mathbb{R}$ and the interval $(0, \infty)$ have the same cardinality.

Prove that Real Numbers and the interval $(0,\infty)$ have the same cardinality. Attempt: Consider the function $f(x) = e^x$. The domain of this function is all real numbers. The range of this ...
1
vote
1answer
27 views

Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$

I want to prove the following equation: $$ f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2}) $$ Is this a valid proof? I am not sure, because at one point I am looking at $f(x) \in ...
1
vote
2answers
28 views

Number of possible unions of a countable number of sets

If $\{ A_{n} \}_{n=1}^{\infty}$ is a countable sequence of distinct sets, then is the number of possible distinct unions between any two or more of the sets in the sequence uncountable? I would like a ...