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
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0answers
37 views

Initial and final topologies

Suppose that $X_i$ are topological spaces, and $X_i \xrightarrow{f_i} Y$ are a family of maps into the set $Y$. The final topology on $Y$ is defined to be the finest topology on $Y$ such that each ...
3
votes
0answers
52 views

Looking for info on power set functor

I was reading here about the various functors which take a set $S$ to its power set. In particular, there is the normal contravariant one, and two covariant ones, which the article calls $\exists$ and ...
0
votes
2answers
36 views

question about proving subset inclusion

normally when one proves subset inclusion, one usually take any $x$ from the subset, and proves that it is also in the superset. e.g. Set $A=$ all triangles Set $B=$ all shapes with a sum of of ...
1
vote
1answer
23 views

I've proved everything about the ideal correspondence easily except $\pi ^{-1} \pi (\frak{a}) = \frak{a}$

The correspondence theorem to which I refer is the bijection between ideals of a commutative ring with $1$, $A$, and ideals of $A/\frak{b}$. I can prove easily most parts that imply the bijection ...
1
vote
1answer
33 views

Decide whether set is convex, connect and bounded.

Let $A=\{ \left(x,y,z \right)\in \mathbb{R}^3 : x^2+y^2-z^2+1<0\}$. Decide whether set A is: a) convex (definition i know: Set $A\in \mathbb{R}^k$ is convex set if for all $x,y \in A$ line segment ...
2
votes
1answer
41 views

Does set theory help understand machine learning or make new machine learning algorithms?

When I was in a university, I didn't major in math but took some math classes. However, I dropped out of math classes pretty quick. Some person recommended that I learn some set theory because it'll ...
-2
votes
1answer
34 views

Operations no Sets

Let $A$ be the set $\{x : x \in \mathbb Z \ \text {and either} \ x ≤ −2 \ \text {or} \ x ≥ 5 \}$ and let $B$ be the set $\{ −3, −2, −1, 4, 5, 6, 7 \}$. Find the following : $A\cup B = \{x : - x ≤ ...
0
votes
4answers
124 views

Is $ (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D)$ true for all sets $A, B, C$ and $D$?

Is $(A \times B) \cup (C \times D) = (A \cup C) \times (B \cup D)$ true for all sets $A, B, C$ and $ D?$ I tried to wrap my head around this, but I have absolutely no idea what is going on here. How ...
-1
votes
1answer
51 views

Are intersection of power set and power set of intersection equal? [duplicate]

Is $P(A) ∩ P(B) = P(A ∩ B)$? At first glance it seems like its not true. I tried writing out all the values of the power set using examples but I'm not sure on how to prove it.
0
votes
2answers
35 views

Prove if the following is true or provide a counterexample if it is not

For all sets A and B, |P(A × B)| $\ne$ |P(A) × P(B)| My first instinct is that it is false and I picked sets like A = {1}, B = {2} but when you write out the power set of these sets you end up with ...
1
vote
3answers
32 views

Is the following set operation true?

Prove the following or else find a counter example: For all sets $A$, $B$, and $C$, $$((A \cup B) − C) \cup (A \cap B) = ((A − B) \cup (B − A)) − C$$ For the life of me, I can't figure out if its ...
2
votes
1answer
15 views

The inverse image of the image of $X$

I'm working on some exercises in Bert Mendelson's Introduction to Topology book in the first chapter and there's this question about functions: If $f:A\rightarrow B$ is injective, then for every ...
0
votes
1answer
35 views

Is $\mathbb{R}^{a\text{ x } a}$ equivalent to $\mathbb{R}^{a^2}$?

In my linear algebra course I was given $\mathbb{R}^{a\text{ x }a} $ as "the set of all $a$ by $a$ matrices". While $\mathbb{R}^{a^2}$ was "the set of vectors with $a^2$ coordinates". Ex ...
-2
votes
0answers
23 views

Proving finite/infinite sets

For j$\in\mathbb{Z}^+$, let $A_j$$\subseteq$$\{$1,..., j$\}$. Suppose that for some n$\in$$\mathbb{Z}^+$, we have B$\subseteq$$\cup^{1}_{j=1}$$A_j$. Is B necessarily finite? Prove it or give a ...
0
votes
1answer
24 views

Find a one-to-one correspondence (i.e, a bijection) [closed]

Find a bijection between the following sets where {[]} denotes a closed interval and {()} denotes an open interval A = [-3,7] and B = [41,100] & A = (-∞,-3) and B = (8,∞)
2
votes
1answer
28 views

Is $\operatorname{card}(I)=\operatorname{card}(D)$

When I was answering number of integrable functions is greater than number of differentiable functions I got to wonder if the inequality was strict. So with $\mathcal I$ being the set of integrable ...
1
vote
2answers
18 views

Prove that the greatest lower bound of $F$ (in the subset partial order) is $\cap F$.

This is one of the question I'm working on: Suppose $A$ is a set, $F \subseteq \mathbb{P(A)}$, and $F \neq \emptyset$. Then prove that the greatest lower bound of $F$ (in the subset partial ...
1
vote
1answer
32 views

Sum of two dedekind cut is a cut

Given $A_1,A_2\in\mathbb R$, define the following: $$ A_1+A_2= \{x + y: x \in A_1, y \in A_2\} $$ I was able to prove that it is not equal to $\mathbb Q$ and isn't the empty set and but I can't prove ...
1
vote
2answers
40 views

What is a good free software to draw complicated Venn diagrams?

The important feature I want is this : I would like to draw two sets as say ovals in solid line but I would like to have the border line in some neighborhood of their two intersection points to be ...
2
votes
1answer
45 views

Is GRP a subcategory of SET, or not? [duplicate]

This is the notion of a subcategory $\mathscr{D}$ of a given category $\mathscr{C}$ which I use: it consists of a subcollection of the collection of objects of $\mathscr{C}$ and a subcollection of the ...
1
vote
1answer
52 views

About subspaces of $\mathbb{R}$ as vector space over $\mathbb{Q}$.

In many texts is noted the analogy between the transcendence degree of a field extension and the dimension of a vector space, so I'm tempting to use such analogy to better understand the structure of ...
1
vote
2answers
70 views

Recursively defining sets of strings discrete math

So here are the two problems: Recursively define the set of bit strings K that do not have 00 as its substring. How many bit strings of length 10 are included in the above set K? Can someone ...
0
votes
1answer
52 views

Consider $f : \mathbb{N} \to (-1,1) \cap \mathbb{Q}, \ n \overset{f}{\mapsto} \sin(n)$. Is $f$ a surjection? If not can we make it a surjection?

This is not homework! I have recently been thinking about the properties of the sine function and whether it can effectively map elements from certain spaces to entirely "fill out" another space. I ...
2
votes
0answers
31 views

A∩B∩C=∅, then A∩B=∅,A∩C=∅ or B∩C=∅ [duplicate]

Sorry everyone, I made a mistake in my question itself for my previous posting: A∩B∩C= ∅. A∩B = ∅, A∩C = ∅ or B∩C = ∅ It should have been: If A∩B∩C=∅, then A∩B=∅,A∩C=∅ or B∩C=∅. Prove or show ...
0
votes
2answers
37 views

Countablity of sets

Why do we choose Natural number to describe whether a set is countable or not? How can we say that Natural Number is countable?
0
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3answers
176 views

Show that $A∩B∩C= ∅$ is only true when $A∩B = ∅, A∩C = ∅$ or $B∩C = ∅$ or show a counterexample.

Show that $A∩B∩C= ∅$ is only true when $A∩B = ∅, A∩C = ∅$ or $B∩C = ∅$ or show a counterexample. My answer: True, Let set A={a,b,c,...}, set B={1,2,3,...} and set C={-1,-2,-3...}. Then there is not ...
0
votes
3answers
45 views

Cantor's diagonal argument and alternate representations of numbers

Cantor's diagonal argument works because it is based on a certain way of representing numbers. Is it obvious that it is not possible to represent real numbers in a different way, that would make it ...
0
votes
1answer
18 views

Express the symmetric difference using only complement and intersect?

How would I go about expressing the symmetric difference if the only symbols I am allowed to use are the complement and intersect symbols? I know that $A \ominus B = (A^c\cap B) \cup (B^c\cap A)$, but ...
-2
votes
0answers
23 views

Recursively defining sets of strings [duplicate]

So here are the two problems: Recursively define the set of bit strings K that do not have 00 as its substring. How many bit strings of length 10 are included in the above set $K$? For (1) I got ...
1
vote
2answers
32 views

Proof by Elements to Show $D^{c} ⊆ A^{c}$

Use proof by elements to verify that for all nonempty sets $A$, $B$, and $D$ if $A ⊆ B$, $D^{c} ⊆ B^{c}$, then $D^{c} ⊆ A^{c}$. Here's the proof I have written so far. I have gotten feedback that ...
-1
votes
1answer
42 views

Binary strings and recurrence relations [closed]

How could we possibly figure this out if there are an infinite number of sequences that could be generated? Could someone please solve this problem and explain what is going on at each step? Thank you ...
1
vote
2answers
16 views

The Null Subset of a Given Defined Set

Given the set $A = \{\{$∅$\},\{2\},2\}$, determine if the following statements are false. If false, then correct the statement to be true Determine the validity of: $\{$∅$,\{$∅$\}\} ⊆ A$ ...
1
vote
2answers
46 views

Construct an explicit bijection $f:[0,1] \to (0,1]$, where $[0,1]$ is the closed interval in $\mathbb R$ and $(0,1]$ is half open.

The problem: Construct an explicit bijection $f:[0,1] \to (0,1]$, where $[0,1]$ is the closed interval in $\mathbb R$ and $(0,1]$ is half open. My Thoughts: I imagine that I am to use the fact that ...
2
votes
1answer
73 views

If there is an injection $f: X \to Y$ with $m=n$ then $f$ is a bijection.

The Statement of the Problem: Let $X,Y$ be finite sets with $ \lvert X \rvert = m $ and $ \lvert Y \rvert = n $. Prove the following statement by induction on $ m \ge 1$: If there is an injection ...
0
votes
1answer
34 views

Proof by Contradiction to show that if $f^{-1}$ exists, $f$ must be onto

Use proof by contradiction to prove that if $f^{-1}$ exists, then $f$ must be onto where $f:A→B$. Proof: I think the contradiction of the theorem would be: if $f$ exists then $f^{-1}$ must be ...
1
vote
1answer
46 views

Showing $(f^{-1}∘g^{-1})=(g∘f)^{-1}$

If the functions $f$ and $g$ are both bijections then the in inverse of the composition function $(f∘g)$ will exist. Show that it will be $(f^{-1}∘g^{-1})=(g∘f)^{-1}$ For the proof assume ...
2
votes
2answers
35 views

Give an example of a set $A$ and a function $f\colon A \to A$ where $f$ is onto but not one-to-one.

I am currently trying to decipher this question but I have been unable to thus far. If a set $A$ is mapped onto itself, it seems that you would always have a function that is both onto and one-to-one. ...
0
votes
0answers
26 views

Proving associativity of symmetric set difference

I'm proving that $P(X)$ (the set of the subsets of $X$) is a ring with the following operations: If $A, B \subset X$, then $A+B := (A \cup B) \backslash (A \cap B) $ and $A \cdot B = A \cap B $. I ...
1
vote
1answer
29 views

Proof that the following integer multiplication is well defined

Prove that multiplication given by $[(a,b)][(c,d)] = [(ac + bd,ad + bc)]$ is well defined. My work: $(a,b) \sim (a_1,b_1) \rightarrow a + b_1 = a_1 + b$ $(c,d) \sim (c_1,d_1) \rightarrow c + d_1 ...
0
votes
1answer
40 views

A doubt about real analysis concerning countable sets [duplicate]

I sometimes get confused when dealing with the notion of countable sets. MY book says that $A$ I countable if there is bijection $f: A \to \mathbb{N }$. but then I frequently see people refer to ...
-1
votes
1answer
20 views

if there is an injection between $A$ and $B$, does there exists an injection between $P(A) $ and $P(B)$?

Suppose $f: A \to B $ is injective, does it follow that there is injective $f :\mathcal{P}(A) \to \mathcal{P}(B) $ ? I mean the obvious choice would be $f( \{ a \} ) = \{ f(a) \} $. does it work ?
0
votes
1answer
34 views

Proving the set identity $(X \cup Y) = X + Y - (X \cap Y)$

I wanted to know if the identity $(X \cup Y) = X + Y - (X \cap Y)$ holds true for any set X and Y? Here, X + Y means all elements in X and Y including repeated elements.
0
votes
2answers
15 views

Trying to find injection between two sets

Let $A,B, C$ be sets such that $f: A \to B $ is injective and $|B| = |C| $. prove there is an injection from $A$ to $C$. Obviously, if we can find and injection $g: B \to C $ then the composition $g ...
1
vote
0answers
57 views

Name of the set $B:= \overline{A}\setminus A$

Let $(X, \mathcal{T}_X)$ denote a topological space and let $A$ be a subset of $X$. We define the set $B:=\overline{A}\setminus A$. Does the set $B$ have a special name in the literature? All I could ...
0
votes
0answers
9 views

Closure systems [duplicate]

Let A be any set. A system $\mathscr{C}$ of subsets of A is said to be a closure system if $\mathscr{C}$ is closed under intersections, i.e. $$\textrm{for any subsystem ...
1
vote
1answer
26 views

Prove if $x$ is greatest lower bound of $U$ then $x$ is the least upper bound of $B$

This is one of the problem I have been solving from Velleman's How to prove book; Suppose $R$ is a partial order on $A$ and $B \subseteq A$. Let $U$ be the ...
1
vote
0answers
51 views

Natural bijection between $\mathbb{N}$ and algebraic numbers?

Q. Is there a canonical, explicit bijection between the natural numbers $\mathbb{N}$ and the algebraic numbers? The earlier MSE question, "Bijection for algebraic numbers," does not quite ...
0
votes
4answers
112 views

If a function maps A to its PowerSet, is it Surjective?

Given an arbitrary set A, let F : A → 2^A be the function defined for all a ∈ A by f(a) = {a} If A maps to its power set, does this make F surjective? If somebody could help to prove this that ...
-1
votes
0answers
38 views

ISO information on powerset functor [closed]

This site has the very bare bones, but I'd like to see more.
6
votes
2answers
114 views

Explicit bijection between $\mathbb Q$ and $\mathbb Z \times \mathbb Z$?

Any idea of an explicit bijection between $\mathbb Q$ and $\mathbb Z \times \mathbb Z$? Even if I think of rational elements as $\frac {m}{n}$, sending them to $(m,n)$ won't work, because all pairs ...