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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1answer
15 views

Can someone tell me how to use basic set notation to say…

A is a set of integers that are not divisible by 3 except the numbers 21, 24, 27, 30, 33, 36, and 39. I'm working on some challenge problems that require set notation and I think I may have it but I ...
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0answers
13 views

Union and Intersection of Indexed Sets

I'm having trouble figuring out how to do this problem in my proofs review. I have a quiz on this tomorrow and I'd really appreciate if someone could help me figure out how to do this problem. ...
0
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1answer
20 views

What do you call the operator that takes in a number and spit out sets?

For example, I want to define an operator $P$ that takes in a value and gives all the partition of that value i.e. $P(3)$ = {{3},{2,1},{1,1,1}} What do I call such an "operator" $P$?
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3answers
28 views

Suppose that $f:X \rightarrow Y$ is surjective and $A \subseteq X$ then $f(X-A) \subseteq Y-f(A)$. True or False?

Suppose that $f:X \rightarrow Y$ is surjective and $A \subseteq X$ then $f(X-A) \subseteq Y-f(A)$. I am supposed to determine whether this statement is true or false. If true I am to prove it. If ...
2
votes
1answer
18 views

Language interpretation dilemma: How do I interpret this textual statement?

How do I interpret this statement? Given a set S containing n real numbers and a real number x, there are two numbers in S whose sum is x. It's not clear to me what we can assume here. I'm not ...
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3answers
30 views

If A and B are subsets of X then $X -(A-B)= (X- A) \cup B$.

If A and B are subsets of X then $X -(A-B)= (X- A) \cup B$. I think this statement is true. I have attempted to started a proof on it and I am stuck. I am in an introduction to proofs class. ...
2
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0answers
25 views

Are there $\omega_1$-many binary sequences with this property?

Is there a family $$\{f_\alpha:\alpha<\kappa\}\subseteq 2^\omega$$ of size $\mathfrak c$ or $\omega_1$ such that for any $n\in\omega$, $\{\alpha_i:i<n\}\subseteq \kappa$, and $n\times n$ binary ...
0
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1answer
31 views

Some basic Mathematical proofs on set theory

After a long time passed from graduation we were recently discussing some matters with my colleagues when one asked me to prove that $ A \cap A = A $ $A \cup A = A$ $\neg (\neg A) = A$ ...
2
votes
1answer
19 views

(Countable) partition generated $\sigma$-algebra

I am working on exercise 1.9 b) in the book "Probability and Stochastics" by Erhan Çınlar to practice my understanding of $\sigma$-algebras (although this might possibly be future homework in my ...
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2answers
28 views

Relation - elementary set theory proof

Claim: Suppose $R \subseteq M x M$ is a relation on a set M such that $R \subseteq R^{-1}$ and R is symmetric .Then $R^{-1} \subseteq R$ also. I had: Suppose R is a relation on set M such that $R ...
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1answer
40 views

Empty Set Proof [duplicate]

Prove that the empty set is a subset of every set. I don't really know where to start other than the fact that I know a symbolic representation of the empty set and that it is included in every ...
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3answers
23 views

Relation - Set theory proof

$When $ A$, B \subseteq \left\{ {1, 2, 3, 4, 5}\right\}. $ $Define $ A $ t $ B to mean that both $\left\vert{A \cup B}\right\vert$ & $\left\vert{A \cap B}\right\vert$ are even. Prove that t is a ...
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2answers
25 views

Subset of Cartesian Product

$A\subset C$ and $B\subset D$ if and only if $A \times B\subset C \times D$. Prove if is correct or incorrect. How to start this, I don't know? is this true or not true?
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2answers
21 views

Set Differences and Union

True or false. Prove why. $A \setminus (B \cup C) = (A \setminus B) \cup (A \setminus C)$ If $x$ is in the right side, then $x$ is in $A$ and not in $B$, or $x$ is in $A$ and not in $C$. $x$ is in ...
0
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2answers
28 views

Proving the inverse of a relation exists

A relation is a set of ordered pairs $(x,y)$ that relates $x$ to $y$ somehow. It's a very weak relation in the sense one thing can be related to many things. A function is a special relation where ...
0
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0answers
22 views

Why do we define the ordered pair in this way? [duplicate]

When we define an ordered pair (x,y) in a set, why, in many textbook, do we define it as {x,{x,y}} or {{x},{x,y}} instead of {x,{y}} or {{x},{y}} which obviously makes more intuitive sense if we ...
0
votes
1answer
22 views

How do the sets have similar properties?

Where they say: "Assume that we have sets $S_k$ with the desired properties for all $k < n$ (line 4 in solution)" What properties are they talking about? They said: "Let $S_n = \{2a ...
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2answers
37 views

The domain onto or into or neither?

My question is actually related to the semantics of onto If $f(x)=\sin(x)$ is $f(x)$ onto the set $[0,1]$? This a question from Real Analysis and Probability by Dudley (page 9). I am confused ...
0
votes
1answer
41 views

Set Theory, working with cardinality, subsets and minimal maximal.

I am asked: Let $A =\{n:n$ is an integer and $-6<n<6\}$ and $X=\{B\subset A : 0<|B|<5\}$. Show that $B_1\le B_2$ if and only if $B_1$ is a subset of $B_2$ defines a partial order on $X$. ...
0
votes
2answers
38 views

Is a vector space with two identical vectors a vector space with one or two vectors?

I'm new this, and cannot find any answers by searching. If a vector space has 2 identical vectors, in particular the zero vector, is it a vector space with 2 vectors or since they are linearly ...
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2answers
34 views

Does it exist a countable set such that any sequence with value in $\mathbb{N}$ is bounded by a sequence in this set?

The question is simple, and the answer may be obvious, but it eludes me... Does it exist a countable set of sequences $S \subset \mathbb{N}^\mathbb{N}$ such that $$\forall v \in ...
0
votes
3answers
77 views

$A ⊂ B$ if and only if $A − B = ∅$

I need to prove that $A ⊂ B$ if and only if $A − B = ∅$. I have the following 'proof': $$ A \subset B \iff A - B = \emptyset$$ $$\implies$$ $$\forall x \in A, x \in B$$ Therefore, $$A - B = ...
2
votes
2answers
280 views

Let A and B be sets, then if A ⊆ B then A ∩ B = A

Let A and B be sets, then if A ⊆ B then A ∩ B = A. How do I prove the above? I'm new to proving, please help! What are the steps I need to create a proof? I'm really lost. Now I know that the above ...
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0answers
25 views

Books of set theory and algebra

Someone knows two good books they focus more on exercises: On following topics: 1) One of set theory and abstract algebra( Groups, ring, modules, ecc..) 2) Another on combinatorics, and discrete ...
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votes
4answers
33 views

Prove that if $A\subseteq B$ then $A\setminus C\subseteq B\setminus C$

How can I prove that if $$A \subseteq B$$ then $$A\setminus C \subseteq B \setminus C$$? I'm a bit lost. I have an idea that follows like so: Let $x \in A, x \notin C$, then $\{x\} \in P(A\setminus ...
2
votes
3answers
177 views

What is the word for set containing other sets?

For example, define $S = \{\{3\}, \{2,1\}, \{1,1,1\}\}$. What is the word for $S$? Multiset? Doesn't seem to be correct. Edit: Also, how can I denote the cardinality of a specific set within a ...
0
votes
3answers
34 views

Proof that sets with 1 element exist from these axioms

I have: Axiom of existence: there exists a set with no elements Axiom of extensionality (equality basically?): if two sets have the same elements they are identical Lemma showing the empty set is ...
2
votes
2answers
37 views

If $A \subsetneq C$ then $A \subsetneq B$ or $B \subsetneq C$. Contrapositive?

If $A \subsetneq C$ then $A \subsetneq B$ or $B \subsetneq C$. Is the contrapositive of this statement If $A \subseteq B$ AND $B \subseteq C$ then $A \subseteq C$. I asked because I think the ...
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2answers
39 views

Why does a set of m elements have 2$^m$ subsets?

Note: This example is from Discrete Mathematics and Its Applications [7th ed, prob 2, pg 576], shout out to @crash. I understand why $A \times A$ has $n^2$ elements(because every member of set $A$ ...
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3answers
49 views

True or False $A - C = B - C $ if and only if $A \cup C = B \cup C$

True or False $A - C = B - C $ if and only if $A \cup C = B \cup C$ I am in an introduction to proofs class. I think this is a true statement. I have began the proof and realize I have to do this ...
2
votes
1answer
32 views

Intersection of family of non negative integers

if $F=\{(-x,x) : x~\text{is a non negative integer} \}$, then what is $\bigcap F$ ? My approach shows that this should be (0,0) = 0, but is this the null set in this context, because the answer is ...
0
votes
1answer
71 views

Is it possible to construct ZFC set theory inside category theory?

It's entirely possible I don't understand what I am talking about, but I know that ZFC stands as a good foundation for much of mathematics and that category theory stands as a good foundation for ...
0
votes
1answer
41 views

Is $\{-{a\over b}\}=\{{a\over b}\}$? (Fractional part)

Given $a,b\in \Bbb{N},(a,b)=1$, is $\{-{a\over b}\}=\{{a\over b}\}$? Or it remains $\{-{a\over b}\}$? The tags I added are the result of me: Not being sure where the notations came from. Assuming ...
4
votes
1answer
37 views

Notation for Average of a Set?

In particular, I have some set $S = \{s_1, s_2, s_3, ..., s_n\}$ and a subset $S^\prime$, and I want to denote the average of the elements in $S^\prime$. I would generally just use ...
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0answers
28 views

Does ℘(A ∩ B)=℘(A) ∩ ℘(B) hold? How to prove it? [duplicate]

I'm currently working on some discrete mathematics work and I've encountered a question I'm not sure how to answer exactly. Precisely, I'm trying to prove that two power, intersected sets statements ...
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votes
2answers
32 views

Show that a closed interval cannot be written as an intersection of finitely many open sets [on hold]

Consider the closed interval [a,b] in R. Show that [a,b] cannot be written as an intersection of finitely many open sets, and then demonstrate that [a,b] can be written as an intersection of ...
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2answers
26 views

order of subgroup same as order of group(finite groups)

If I have order of a subgroup C of same order as group G I want to prove that G = C. One inclusion is obvious C $\subset$ G the other inclusion we can get by a bijection f : G $\rightarrow$ C hence ...
0
votes
1answer
30 views

Notation about sets and probability

Let $p \to \left [ 0,1 \right ]$ be a function and $X$ a pointprocess on $S=\mathbb{R}^d$. Then \begin{align} X_{\mathrm{thin}} = \lbrace u\in X \mid R(u)\leq p(u) \rbrace \subseteq X ...
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2answers
50 views

One to One vs. Onto

Is there a one to one function mapping the positive integers to the open interval (0,1)? Is there an onto function mapping the positive integers onto the open interval (0,1)? I'm having trouble ...
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votes
0answers
24 views

show that for $x\in I$, the principal filter $D=\{X\subseteq I:x\in X\}$ is a filter on$I$ [on hold]

1)show that for $x\in I$, the principal filter $D=\{X\subseteq I:x\in X\}$ is a filter on$I$. 2)show that every principal filter is an ultrafilter
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votes
7answers
771 views

Is an empty set equal to another empty set? [duplicate]

I have a definition that claims that two sets are equal A = B, if and only if: $\forall x ( x \in A \leftrightarrow x \in B)$ An empty set contains no elements. If I define the sets: A = ...
0
votes
2answers
40 views

Union of non-intersecting finite sets

Suppose that $|A|=k$ and $|C|=n$, finite cardinalities, and $A\cap C= \varnothing$. What is $|A\cup C|$? Prove your answer. I have done problems similar before, but none with finite cardinalities, so ...
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3answers
43 views

How to visualize the Cartesian product of a set and an interval

For example, suppose I have the following: $[1,5] \times \{-2,2\}$. I know that $[1,5]$ is an interval and the set $\{-2,2\}$ is a set containing two elements, but I am getting confused with the ...
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0answers
24 views

A Question on Continuous Functions over Topological Spaces

Let $f:X \rightarrow X$ be a continuous function on a topological space $X$. Under what conditions is it the case for every subset $A \subseteq X$ that $$A \cap \bigcap_{i=1}^{\infty} f^{-i}((X-A)\cup ...
3
votes
1answer
124 views

Which areas of modern mathematics don't use ZFC set theory?

I've heard that Algebraic Geometry requires something called Category Theory, which itself requires an extension of ZFC called Tarski-Grothendieck set theory, and that got me wondering. Which areas ...
0
votes
1answer
50 views

Is there a Set Theory textbook which include visual explanation?

Right now I am taking a class in set theory, The professor in many cases draw a diagram,pic or anything which help you intuitionally understand the material better. What was surprised me many (at ...
0
votes
1answer
36 views

$\operatorname{Card}(X) \leq\operatorname{Card}(Y)$ iff $\aleph (X) \leq \aleph(Y)$

For any two sets $X$ and $Y$, we write $\operatorname{Card}(X)\leq\operatorname{Card}(Y)$ if an injection $X \rightarrow Y$ exist. I have tried Suppose $\aleph (X) \leq \aleph(Y)$, where $\aleph ...
0
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2answers
30 views

Show that $B((a,b),r) \subset A\times B$

Question: Let $A$ and $B$ be non-empty subsets of $\mathbb{R}$. Prove that $A \times B$ is an open subset of $\mathbb{R}^2$. NOTE: Fellow M.SE users pointed out that the question lacks ...
0
votes
2answers
30 views

Set difference is distributive over intersection proof?

I'm asked to prove the following: $(A-B) ∩ C = (A ∩ C) - (B ∩ C)$ using the definitions of intersection and difference. So far I have: $(A-B) ∩ C = \{(x ∈ A) ∧ (x ∉ B)\} ∧ \{(x ∈ C)\}$ "Defn ...
1
vote
3answers
38 views

Set cardinality, function onto, open unit square maps into real number set

I have a question in my homework that I have trouble solving it. I'm not sure if I understand the question actually. I'll attach the question below and hope someone could give me any hints. Consider ...