This tag is for elementary questions on set theory - the sort of material covered in "Chapter 0" of graduate texts and in undergraduate set theory texts. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, countability and uncountability, power sets, ...

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1answer
204 views

Venn diagram related question

An analysis of the survey of $320$ school pupils highlighted the following facts: • $50$ pupils live in New Town, travel to school by bus and have canteen lunch. • $110$ pupils live in New Town ...
2
votes
3answers
591 views

Image of a union of collection of sets as union of the images

I am having problems with establishing the following basic result. Actually, I found a previous post that is close in nature (it is about inverse image), but I was interested in this specific one, ...
3
votes
2answers
710 views

Study of Set theory: Book recommendations?

Can you suggest a good book for set theory? I have just started reading about Group theory and want to learn set theory on my own. Thanks in advance
1
vote
1answer
45 views

strict ordering a Set

Given: $(A,<)$ is a strictly ordered set and $b \notin A$. Define: a relation $\prec$ in $ B = A \bigcup \{b\} $ as: $x\prec y $ if and only if $(x,y \in A \text{ and } x<y)$ or $(x \in A ...
2
votes
2answers
84 views

Equivalence Relations and functions on partitions of Sets

let $f$ be a function on $A$ onto $B$. Define an equivalence relation $E$ in $A$ by: $aEb$ if and only if $f(a)=f(b)$. Define a function $\phi$ on $A/E$ by $\phi([a]_{E})=f(a)$. Hint: Verify that ...
4
votes
1answer
214 views

Set Theory Notation Crises

For those who are familiar with the following notation, could you explain it in plain English because I picked up a set theory textbook but the book assumes the reader is familiar with the notation ...
0
votes
2answers
49 views

$[(|X| \ge \aleph_0) \;\wedge\; (A \subset X) \;\wedge\; (|A| = |X\backslash A|)] \Rightarrow |A| = |X|$

Suppose that $X$ is an infinite set of cardinality $\alpha$. Also, suppose that, for some $A \subseteq X$, we have that $|A| = |X\backslash A|$. I want to show that $|A| = |X|$. When, for example, ...
0
votes
2answers
113 views

generalizing De Morgan's Laws [duplicate]

Show: $$ B - \bigcup_{a \in A}F_{a} = \bigcap_{a \in A} (B - F_{a}) $$ and show $$ B - \bigcap_{a \in A}F_{a} = \bigcup_{a \in A} (B - F_{a}) $$ I struggle with proofs. This is what I have for the ...
4
votes
4answers
243 views

Does every ordinal have cardinality no greater than $\aleph_\mathbb{0}$?

My notes say that the ordinals $\omega + 1, \omega + 2, ... , 2 \omega, ... , 3 \omega, ... \omega^2, ... $ are all countable, and hence have cardinality equal to $\omega = \aleph_\mathbb{0}$. So I ...
4
votes
1answer
129 views

Set Notation (Axiom of Replacement)

This question is related to the one I asked yesterday here in that it's related to another one of the Zermelo-Fraenkel Axioms. After looking over the notation used to describe the axiom, that is: $$ ...
1
vote
1answer
77 views

functions and infinite intersections

let $f$ be a function. If it is given that $$ f\left[\bigcap_{a\in A}F_{a}\right]\subseteq\bigcap_{a\in A}f[F_{a}]$$ then, if it is further given that $f$ is one to one, prove that the $\subseteq$ ...
2
votes
0answers
67 views

“Let A be a set. We are able to quotient all possible well-orders over the set A.” What does this mean?

"Let A be a set. We are able to quotient all possible well-orders over the set A." This was the first line in the set-up of some exercises I have to do (which ask specific questions depending on ...
3
votes
2answers
71 views

When does equality holds in $A\subseteq P(\cup A)$

Note: $P$ is power set. It's easy to prove that this inclusion holds. But when is other inclusion true? I can't even think of one example...
2
votes
3answers
121 views

The difference between $\mathbb{Z}$ and $\mathbb{Z}^2$

I know that $\mathbb{Z}$ is the set of integers. But, what does $\mathbb{Z}^2$ mean? How is it different from $\mathbb{Z}$? Thanks.
3
votes
2answers
103 views

Set Notation (Axiom of Infinity)

I'm having trouble understanding the notation used in describing the axiom of infinity (which is number 6 in the Wolfram MathWorld page). I understand what the axiom is saying, but I'm trying to ...
5
votes
2answers
176 views

Valid Proof that the Irrationals are Uncountable?

So I originally wanted to prove that the reals are uncountable, but the best solution I came up with was to prove the irrationals are uncountable so therefore the reals must be as well. I suppose my ...
0
votes
1answer
49 views

Need help in finding counterexample

I need to find example that this isn't correct: Let $R_1,R_2,R_3$ be binary relations on set $A$. Prove that this is not correct: $(R_1\cup R_2)\circ R_3 \supseteq(R_1\circ R_3)\cup(R_2\circ R_3)$
1
vote
2answers
81 views

Defining a subset

The question I have to answer is as following in Swedish: Hur många mängder X uppfyller {a, b, c} ⊆ X ⊆ {a, b, c, d, e}? Loosely translated (I do not know ...
0
votes
1answer
43 views

show that there exists a set with $x_{1},…,x_{n}$ as its elements

Let $x_{1},....,x_{n}$ be sets for $n\in\mathbb{N}$ and $n\ge 1$. a) show that there exist a set with $x_{1},....,x_{n}$ as its elements b)show that $x_{1}\cup x_{2}\cup...\cup x_{n}$ is a set. My ...
0
votes
1answer
37 views

infinite intersection of sets

I'm trying to proof this identity: $ [a,b]\equiv \bigcap_{n=1}^\infty [a,b+\frac{1}{n}) \equiv \bigcap_{n=1}^\infty (a-\frac{1}{n} , b] $ I already try to use De-Morgan's lows but with no success . ...
2
votes
1answer
60 views

How to construct a function from a pair of possibly empty sets?

I am stuck on an elementary proof on the cardinality of sets on the following point: Given two possibly empty sets, $A$ and $B$, I need to prove the existence of any function $f:A\rightarrow B$. Is ...
1
vote
1answer
42 views

Specific element in a set

Given a set $A=\{a,b,c,d\}$ how can I take, for an example, the first element? Like this: $A(0)=a; A(1)=b; A(0).A(1)=a.b$. I think there's no way to do that with sets, because it isn't an ordered ...
5
votes
1answer
101 views

Does the definition of countable ordinals require the power set axiom?

I am trying to understand the consequences of the different axioms of ZFC. In particular, I was trying to understand what you get on ZFC-power set (ZFC minus the power set axiom). If you have any ...
0
votes
3answers
196 views

Can countability coexist with infinity?

This question concerns the countability of the real numbers. First I will show how I count the numbers between 0 and 1 on the real line. It is done by reversing digits behind the coma, so that e.g. ...
3
votes
4answers
120 views

Set that is not well-ordered

I want to find a set that is transitive, every non-empty subset of it has $\in$-maximum and is not $\in$-well ordered. Any hint?
0
votes
1answer
187 views

Metric and the triangle inequality

Let $A,B$ be finite subsets of the natural numbers. If we let $d(A,B)=\sum_{x\in A\mathbin\Delta B} 2^{-x}$, where $A\mathbin\Delta B=(A\cup B)\setminus (A\cap B)$ is the symmetric difference between ...
1
vote
3answers
84 views

Properties of basic set theory

The question is about a set: $$B=\{a_1,a_2,a_3,...,a_n\} \subset \mathbb R$$ And would like to know how to calculate $B^n$ where $n \in\Bbb N$?
0
votes
1answer
98 views

The set of all functions mapping set $A$ to set $B$

Is $F$ as defined here the set of all functions from set $A$ to set $B$? $F=\{f\in 2^{A\times B}:\forall x(x\in A\rightarrow\exists y (y\in B\wedge (x,y)\in f))\wedge \forall x,y_1,y_2 ((x,y_1)\in ...
11
votes
3answers
2k views

Cardinality of the set of prime numbers

It was proved by Euclid that there are infinitely many primes. But what is the cardinality of the set of prime numbers ? Cantor showed that the sets $\mathbb{Q}$ and $\mathbb{Z}$ have the same ...
2
votes
4answers
814 views

Transitivity of union of two transitive relations

I have a question concerning proving properties of Relations. The question is this: How would I go about proving that, if R and S (R and S both being different Relations) are transitive, then R union ...
2
votes
2answers
336 views

how many empty sets are there?

Would I be correct in saying that in the category of sets, the "class of sets that are isomorphic to the empty set is a proper class"? In other words, there are LOTS of initial objects in the ...
1
vote
1answer
109 views

Finite ring of sets

I have some questions about finite rings of sets and I'll be very grateful for any help. Let E be some fixed non-empty set. Suppose we are given some finite ring of subsets of set E, i.e. some ...
0
votes
1answer
125 views

show that a set called ordered pair is unique

For any sets $x$, $y$ we write $\langle x,y \rangle$ for the set $\{\{x\},\{x,y\}\}$ Show that this set is unique. My proof: By axiom ZF3 of pairs there exists a set (I call it $z:=\langle x,y ...
7
votes
7answers
379 views

Rational numbers $\mathbb Q$

$$\Bbb{Q} = \left\{\frac ab \mid \text{$a$ and $b$ are integers and $b \ne 0$} \right\}$$ In other words, a rational number is a number that can be written as one integer over another. ...
1
vote
1answer
1k views

cardinality of a set with repeating elements? [duplicate]

What is the cardinality of a set which has repeating elements ? For example $S = \{1,1,1,2,2\}$ Is each individual element counted? Please quote a reference text if possible.
7
votes
2answers
92 views

Closed curves in 3D

Is $R^3$ a union of disjoint closed curves? (Obviously $R^3$ minus a line is). Is this a classical problem?
4
votes
4answers
250 views

Intersection of collection of sets not in the collection of sets?

Hello I was doing a practice problem and I came across something that kind of tripped me up. The question says: Suppose the Collection $F$ is given by $F=\{ [1, 1+ 1/n] : n\in \mathbb{N}\}$ Find the ...
1
vote
5answers
221 views

cardinality of the union of 2 sets with at least two elements and cardinality of cartesian product

Suppose X,Y are sets with at least 2 elements. Show that $X\cup Y\le X\times Y$ So my first thought was that cardinality $|X|\ge 2$ and the same for $|Y|\ge 2$ but by the inclusion-exclusion ...
1
vote
2answers
69 views

cardinality problem concerning inclusion of sets

if $A\subseteq B$ then $A\le B$ , i,e if A is a subset of B then A is less than or equinumerous with B How to do that? IF A is a subset of B then every element in A is an element in B, so A cannot ...
0
votes
2answers
280 views

Shading A Venn Diagram Using A Specific Equation

The expression is: $A\triangle(B\cap C')$. The $\triangle$ refers to $-$ and the $\cap$ refers to an intersection, whilst the $\;{}'$ refers to the prime of $C$. There are no numbers or items ...
1
vote
1answer
48 views

question about sets

I have this as a beggining to a question: $A\subseteq Z^2$ $$ A = \left \langle \left ( 1,7 \right );\left ( 7,2 \right );(2,3) \right \rangle = \left \{ ...
1
vote
1answer
60 views

is this function injective?

Is this function injective? $f:\mathbb{N}\to \mathbb{N\times N}$ defined as $f:n\to (n, n+1)$ $f(n_{1})=f(n_{2})\Rightarrow (n_{1},n_{1}+1)=(n_{2},n_{2}+1)\Rightarrow$ $n_{1}=n_{2} \wedge ...
2
votes
3answers
3k views

Set Distributive Property Proof

Prove the distributive property for sets: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ I'm not good with proofs but my understanding is that I have to prove 2 things: (1) $A \cup (B ...
0
votes
1answer
242 views

Showing that a union of the subsets of two $\sigma$-algebras is a $\sigma$-algebra.

I got back an assignment for a first course in analysis and I have made a very basic error, and I'm having a lot of trouble pinpointing exactly what piece of information I'm missing. You have two ...
0
votes
2answers
93 views

Prove by induction that $n^{+}+m=n+m^{+}$ for all $m,n \in \omega$

Just to make sure we use same symbols: $\omega= \mathbb{N}_{0}$, $n^{+}$ is successor of number $n$. And we define addition by recursion: $+:\omega \times \omega \rightarrow\omega$ $m+0=m$ ...
3
votes
2answers
71 views

Why Zorn's Lemma “fails” on nested boxes problem?

Backgound My problem comes from the nested boxes problem. Consider a list of boxes. Each box has a length, width, and height. Since the boxes can be rotated those terms are interchangeable. The boxes ...
3
votes
1answer
141 views

Can we use proper classes in this way, to define a new infinity larger than |Ord|?

I believe there is a way to do this that makes sense, and I explain it below. I would like to know if I did some obvious mistake, or if the idea doesn't make sense for some reason I didn't figure it ...
2
votes
1answer
94 views

Set Theory: show that function h is injective

If $h$ belongs to the set of functions from $X$ to $X$, why is $h$ injective if and only if, for any functions $f$ and $g$ in that set, if $h \circ f = h \circ g$, then $f=g$? I'm trying to ...
5
votes
2answers
166 views

Showing ${\frak c}:=|\Bbb{R}|=2^{\aleph_0}$ using only the axioms of the reals

It is a well-known fact that the cardinality of the continuum, ${\frak c}:=|\Bbb{R}|$, is the same as the cardinality of the powerset of the natural numbers, but I've tasked myself with proving the ...
5
votes
2answers
346 views

Subtracting two infinities

I am Curious if the following is mathematically correct: Let $a$ be the infinite set of all nonnegative integers $0,1,2,3...$. I take from $a$ some of its elements, say integers $10$, $11$, and $12$ ...