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
46 views

Countably infinite and monotonically countably infinite

While reviewing Meaning of finite, countably finite, infinite?, I wondered about $\mathbb N$ vs $\mathbb Q$, both countably infinite, but $|\mathbb Q|$ is clearly much nearer to $|\mathbb R|$ than ...
2
votes
1answer
49 views

Countable and uncountable set.

Which of the fallowing sets of functions are uncountable? $\{f|f:\Bbb N\to\{1,2\}\}$ $\{f|f:\{1,2\}\to\Bbb N\}$ $\{f|f:\{1,2\}\to\Bbb N, f(1)\le f(2)\}$ $\{f|f:\Bbb N\to\{1,2\}, f(1)\le f(2)\}$ I ...
1
vote
4answers
300 views

Meaning of finite, countably infinite, infinite?

Even after several attempts I could not find the motivation behind the finite, countable and infinite. Is there a simple way to look them differently? I have read the wikipedia definition several ...
5
votes
5answers
249 views

Theoretical function question

Suppose we have the function $f(x)= x^2 $. This function associates real numbers with real numbers ( $f:\mathbb{R}\rightarrow \mathbb{R}$). Now, what i get confused sometimes is what exactly the ...
1
vote
2answers
27 views

Axiom of separation for $n$ tuples or $n$ place predicates

The axiom of separation seems to only work when you are using an arity 1 type predicate, how then can we form relations? I know the power axiom allows for you to work with a set of subsets and in turn ...
2
votes
4answers
30 views

Proof of theorem about equivalence classes

I am looking to understand the following theorem, and I am also wondering what is meant by "mutually disjoint", or at least how it's to be understood in the following context: The distinct ...
0
votes
3answers
52 views

Can someone explain what this theorem and proof is saying

can someone please explain what the following theorem and proof is saying. Thanks in advance
2
votes
1answer
46 views

Prove $((A^C \cup B^C) \setminus A)^C = A$

I have attempted this proof as outlined below. However, I feel that it is not correct, any suggestions would be appreciated. Prove $((A^C \cup B^C) \setminus A)^C = A$ L.H.S. $((A^C \cup B^C) ...
3
votes
3answers
49 views

Why isn't the empty set an element of $A \times B$, while it is a relation from $A$ to $B$?

Let $A$ be $\{1,2\}$, let $B$ be $\{x,y\}$. According to the information I get from most textbooks, $$A \times B = \{(a,b): a\in A\text{ and } b\in B\}$$ $$A \times B = ...
1
vote
2answers
52 views

Trying to understand a proof about onto/1-1 mappings (from Herstein's Topics in Algebra)

I am working on some problems in a book I have and I want to make sure that I have an accurate possible proof. That is, I want to make sure I actually understand/ can justify the reasoning. (some of ...
0
votes
1answer
29 views

If a function $f$ is invertible can I say that $f^{-1}$ is also one to one and onto?

If we have a function $f$ that is both one-to-one and onto (so it's invertible). Its inverse function $f^{-1}$ is also one-to-one and onto? If this is not true can someone please explain it to me or ...
-1
votes
2answers
48 views

Guarantee that the intersection of a family of sets satisfies $P$ when every member satisfies $P$

1) Suppose we are given a family of sets $\{A_\alpha\}_{\alpha\in\Lambda}$ indexed by a set $\Lambda$ of arbitrary cardinal. Suppose that in addition the sets $A_\alpha$ are endowed with some property ...
2
votes
1answer
29 views

Determining if two sets are equal, subsets of one another, or neither

Problems: 1.) $A = \{x \mid x^4 - 3x^2 = 4\}$, $B = \{x \mid x^2 - 4 = 0\}$ 2.) $A = \{x \in \mathbb{C} \mid x^3 = 1\}$, $B = \{x \in \mathbb{C} \mid x^2 + x +1 = 0\}$ 3.) $A$ = The (real) domain ...
0
votes
0answers
27 views

Addition set confusion [duplicate]

Prove that for any two sets we have the following: ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣. Not sure what the + means here.
1
vote
1answer
32 views

The Cantor set and ternary expansions

I'm trying to prove that the Cantor set $\mathcal{C}$ contains all numbers $x \in [0,1]$ with ternary expansion $x = \sum_{k=1}^\infty \frac{a_k}{3^k}$, such that $a_k=0$ or $a_k=2$. I'm going by ...
0
votes
1answer
47 views

Set Addition proof

Prove that for any two sets we have the following: A∪B=A+B−A∩B Not sure what the + means here. Do i just add the two sets together including the duplicated elements? Yes sorry it should be ...
1
vote
1answer
21 views

Are two segments order isomorphic

I thought $[0,2]$ and $[0,1] \bigcup (2,3]$ are order isomorphic, since I can write the isomorphism $f:[0,2]\rightarrow [0,1] \bigcup (2,3]$ : $f(x)=x $ when $x\in [0,1]$ $f(x)=x+1$ when $x\in ...
0
votes
2answers
41 views

Show that every non-empty set of integers that is bounded below has a minimum.

In a real analysis problem, I assumed directly that for all the integers greater than a real number, there is an integer closest to that number. I want to somehow justify this. Is this a ...
0
votes
0answers
30 views

Set Theory text with solutions to exercises [duplicate]

I'm looking for a set theory text that has solutions to the exercises. I will be studying on my own and want to be able to check my understanding. Thanks for the suggestions.
1
vote
2answers
40 views

Show a set is a closed set larger than open set

let $(\mathbb{R}^n, \tau)$ where $\tau$ is the std topology. Show that the set $U = D_{\delta}(x) = \{y\in \mathbb{R}^n: ||y-x||\leq \delta \}$ is a closed set. so under these conditions it means I ...
3
votes
0answers
49 views

Why is this not a proof of Schroeder-Bernstein?

We can show that if $f: A \rightarrow B$ is injective then $|A| \leq |B|$ and if $g: B \rightarrow A$ is injective then $|B| \leq |A|$ so $|A| = |B|$. By the definition of having equal cardinality, ...
2
votes
2answers
94 views

Question about set of all functions and the power set of a set

Hi so we know the set of all functions from a set X $\phi \rightarrow$ {0,1} create a one to one correspondence from the power set of X to the set of all functions but we are looking at certain ...
0
votes
2answers
42 views

A point is in $\partial A$ iff every neighborhood of it contains both a point of $A$ and a point of $X\setminus A$.

How to prove (c) A point is in $\partial A$ iff every neighborhood of it contains both a point of $A$ and a point of $X\setminus A$. (d) A point is in $\operatorname{cl}(A)$ if and only if ...
0
votes
0answers
107 views

What to teach in Set Theory & Logic Course. [migrated]

I will be teaching a third-year introductory course on Set Theory and Logic soon and was hoping to get advice from this community. I would rate my students' proof abilities as weak and was hoping to ...
3
votes
1answer
57 views

Proof that every field $F$ has an algebraic closure $\bar F$

I am reading the book A First Course in Abstract Algebra written by Fraleigh and I do not really understand the proof of theorem 31.22, that every field $F$ has and algebraic closure $\bar F$. I ...
2
votes
7answers
96 views

Any hint in how to simplify a set theory expression: $(A \cap B) \cup (A \cap B \cap C' \cap D) \cup (A'\cap B)$

I'm having trouble simplifying this set theory expression $$\begin{align} (A \cap B) \cup (A \cap B \cap C' \cap D) \cup (A'\cap B) & \end{align} $$ In the books says that absorption law can ...
1
vote
1answer
38 views

Exclusive OR a set with itself

Given set A is a finite set, then $ A\oplus A=\emptyset $ and $A\oplus \emptyset = A$. These make perfect sense to me, since the XOR operator requires only one "True" Condition for the output to be ...
1
vote
0answers
38 views

Feedback on Set Theory Questions - Study Help

I am doing some previous exams on Discrete Math as practice and I am just looking for some feedback on the answers as I don't think they are available publicly. I am reasonably confident on some, but ...
2
votes
2answers
43 views

A bit confused about definition of set of mappings in Herstein's Algebra

I have been just trying to start working with the books Topics in Algebra by I.N. Herstein, and I am having a bit of trouble understanding a definition. It is, Definition: If $S$ is a nonempty ...
2
votes
2answers
46 views

Every subset has first and last element -> set is finite

Let $X$ be a partially ordered set, so that every non empty subset of $X$ has a first and a last element. Show that $X$ is a finite set. And what if every subset only has a first element? Well, I ...
2
votes
1answer
40 views

Convert the given set into roster notation:

Problems: Find $A$, where 1) $A = \{x \in \mathbb{R} \mid x^4 - 1 = 0\}$ 2) $A = \{x \in \mathbb{C} \mid x^4 - 1 = 0\}$ Attempt: Solving the equation: $x^4 - 1 = (x^2)^2 - 1^2 $ $= (x^2 - ...
0
votes
0answers
31 views

Measure Theory by Halmos, theorem B, page 22

This is an excerpt from Halmos' "Measure Theory" at page 22 and 23. This proof seems wrong to me. I could not prove that the class of all finite union of elements of $E$ is a ring of sets. I can only ...
1
vote
3answers
31 views

Disjoint refinement of a set of sets

Say I have some set $F$ of sets. This is obviously a cover of $\cup F$. Is there a general algorithm that makes "this" cover disjoint? That is, a set $F'$ of pairwise disjoint sets with the properties ...
2
votes
3answers
40 views

Discrete Math: Unions, Intersections, Complements

Are these answers correct? The union and intersection only include the elements in the universal set? $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ (where $U$ is only a subset of the Universe) $A ...
3
votes
1answer
51 views

What are the bounds (upper and lower) for $|A+A|$?

Let $A$ be a finite set of real (or complex) numbers. If I consider sets with small sizes, we have that: If $A$ is the empty set, then $A+A$ is also empty. If $A$ is a singleton, then $A+A$ is ...
1
vote
0answers
30 views

What is meaning of big U in sets? [duplicate]

What does big U below signify? And what is number written above and below it?
3
votes
2answers
21 views

Computing the union and intersection of family of sets

Suppose we are given for all $n \in \mathbb{N} $ $$ X_n = \{ (x,y) \in \mathbb{R} \times \mathbb{R} : n^2 \leq x^2 + y^2 \leq (n+1)^2 \} $$ I am trying to compute $\bigcup_{n \in \mathbb{N} } X_n $ ...
1
vote
1answer
29 views

Inverse Function in terms of Surjective and Injective Functions

Here is my intuition of the proof listed after The mapping of A to A is an inverse function. The mapping of A to B is injective and the mapping from B to A is surjective. I'm confused as to where ...
2
votes
1answer
29 views

Find the number of times K appears in any 4 item subset of T

Given the set T of all K {1, 2, 3, 4, 5, 6, 7, 8, 9} Let N be 4. There can be produced 126 combinations of N items, as subsets S. Every K has an equal ...
0
votes
2answers
17 views

Trying to prove a set equality

Suppose $f:X \to Y $ is a map and let $A \subseteq X $ And $T \subseteq Y $. I want to show that $f^{-1} ( f(A)) \supseteq A $. Also, I want to show that $T = f( f^{-1} (T)) \iff T \subseteq Im f $. ...
1
vote
1answer
47 views

A question on the generalization of Cartesian Product

In Halmos’s Book, it is written that, The notation of families is the one normally used in generalizing the concept of Cartesian product. The Cartesian product of two sets $X$ and $Y$ was defined ...
0
votes
1answer
32 views

Set operations on $A$ and $B$

I am a bit confused right now, in one of the practice questions for my book it says $$A,B \subset X$$ $$B = A \cup ((X \setminus A) \cap B)$$ However, when I simplify it, I get that it equals $$A ...
1
vote
1answer
23 views

Question about methods of proof (Elementary Set Theory)

I have a question in regard to some questions I am working on in beginner set theory. I will give an example to illustrate by question better, For example, say we were wanting to prove $$A \cup(B ...
1
vote
1answer
21 views

$S(\Omega \sqcap A)=S(\Omega)\sqcap A$ Halmos Measure Theory

I'm having trouble grasping the proof of theorem E, section 5, chapter 1 in Halmos' Measure Theory. Let $X$ be a nonempty set, and $\Omega$ a family of subsets of of $X$. Given $A\subset X$, denote ...
2
votes
1answer
18 views

Proving or Disproving a function that is onto itself is one to one.

I'm having some trouble formulating a proof for this following problem: A is a finite set and f a function with f : X → X. Suppose that f is onto. Now Prove or Disprove: f is one to one. ...
0
votes
2answers
33 views

Set theory basics exam

I have a question about this, we had this on our exam. Let be $f:A \to B$ a function. Prove next statements or give an example against it. (i): if $A$ is countable, then $f(A)$ is also ...
1
vote
2answers
42 views

Examples of non-transitive sets.

What are some examples of non-transitive sets? I have conducted several searches on Google and also searched the math.stackexchange website. I have encountered intransitive sets before but cannot ...
0
votes
1answer
125 views

Set theory identity, can't derive it

I see the following step in a textbook, and I can't follow it, $A\cup(B\cup(C-D)-E)$ $=$ $(A\cup(B-E))\cup(C-(D\cup E)$ My progress from the LHS get stuck like this, $$A\cup(B\cup(C-D)-E)$$ ...
2
votes
0answers
35 views

Proof of a lemma needed for proving a second result

So i have a question that i have to prove, but in order to prove it i need to prove the following lemma. It is a typical set theory sort of lemma, so i feel the proofs are almost complete, but ...
1
vote
1answer
38 views

Preimage of a closed set is a closed implies f is continuous. Some concerns about the proof

Ok. I have managed thru: If f is continuous then the preimage of open set is a open set If the preimage of the open set is a open set then f is continuous If f is continuous then the preimage of a ...