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
45 views

Equivalent Metric Using Clopen Sets

Prove that if $(X,d)$ is a metric space and $C$ and $X \setminus C$ are nonempty clopen sets, then there is an equivalent metric $\rho$ on $X$ such that $\forall a \in C, \quad \forall b \in X ...
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1answer
61 views

Why does this author define cardinality indirectly?

I'm studying Enderton's Elements of Set Theory and in the page 129 he defines what it means two sets being equinumerous: After that in the page 136 he defines cardinality: Why doesn't he define ...
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3answers
22 views

Set Theory Proof with Complements

If $A \cap B = \emptyset$ then $A \subset B'$ and $B \subset A'$, where the prime symbol denotes the complement of each set. Here are my thoughts: Assume $A \cap B = \emptyset,$ since the ...
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1answer
35 views

On the good set principle and sigma fields.

Following Probability and measure Theory by Ash (2000). let $\Omega$ be a set, let $C$ be a class of subsets of $\Omega$ and $A \subset \Omega$, we denote by $C \cap A$ the class $\{ B \cap A : B \in ...
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1answer
38 views

Example that the union of sigma algebra is not an algebra

I've tried to find the one, but failed to solve it. Some people asked similar question, but all the answers were about the case that "the union of sigma-algebra is not a 'sigma-algebra'". What I ...
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2answers
49 views

Prove that if $A \subset B$ then $P(A) \leq P(B)$

I'm supposed to prove that if $A \subset B$, then $P(A) \leq P(B)$. The hint it gives is confusing me even more. It says use a venn diagram to convince yourself $ B = A \cup (A^c \cap B)$ and $A$ and ...
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3answers
38 views

Understanding basic intersection and union formula in statistics

I'm having some trouble really understanding this formula when applied to a problem: $ P(A \cap B) = P(A) + P(B) - P(A \cup B) $ The problem I'm given is this: Suppose we draw one card at random ...
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1answer
35 views

Very simple question about subsets

Is {1, 2, 2, 3} a subset of {1, 2, 3} because all of the elements in {1, 2, 2, 3} are contained in {1, 2, 3}? However, {1, 2, 2, 3} isn't part of the power set of {1, 2, 3}, right? Thanks!
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1answer
18 views

Find if relation is partial / total ordered

These are two problems of the Velleman's How to prove book in which it has been asked to find if $R$ is an partial order or/and an total order: $R1 = \{(x,y) \in A ...
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1answer
35 views

How do I prove cardinality is well-defined?

I define equinumerous and cardinality in this way: $A$ and $B$ are equinumerous (written $A\sim B$) if there is a bijection between them. We say $card(X)=card(Y)$ if $X\sim Y$. I would ...
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0answers
49 views

Contructive implications of Cantor's diagonalisation argument [closed]

Following the question here (which is not duplidate), can the argument of Cantor (the famous diagonalisation argument that $\mathbb{R}$ is not countable) ever be constructed (or be finished)? ...
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3answers
76 views

Prove that if $A$ and $B$ are measurable, then $\lambda(A)+\lambda(B)=\lambda(A \cup B)+\lambda(A \cap B)$

Prove that if $A$ and $B$ are measurable, then $\lambda(A)+\lambda(B)=\lambda(A \cup B)+\lambda(A \cap B)$ I tried to prove it using $A\cup B =(A\cap B^c)\cup (A\cap B)\cup(A^c\cap B)$ but ...
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0answers
35 views

Set-theoretic difference between cartesian products as a disjoint union

Let $m$ be any positive natural number and let $A_1$, $C_1$, ..., $A_m$, $C_m$ be arbitrary sets. Is this correct? $(A_1 \times A_2 \times A_3 \times A_4 \times ... \times A_{m-1} \times A_m) ...
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0answers
28 views

Can you name a set of criteria if a number of sets satisfy the criteria?

I'm trying to set out a definition that involves a function from any set that meets these criteria: $ \langle p| p\in\mathbb{R}\ \land p \geq0 \, \, \land \sum p= 1 \rangle$ (I've only used sigma ...
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0answers
13 views

Is there a default evaluation order for the set difference operator?

I know that the set difference operator ('\') is not associative, but I was wondering whether or not by convention it has a default evaluation order specified for it. That is, is it considered to be ...
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2answers
86 views

Diagonalizing a power set

$S$ be any non-empty set, $2^S$ denote the power set of $S$. Let $f$ be a function from $S$ to $2^S$, where for each $x \in S$, $f(x) \subseteq S$. Also, $f$ is injective. Show that $f$ cannot ...
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2answers
42 views

What is the word for the two numbers in a set with the largest range?

For the set $\{1, 2, 3, 4\}$, $1$ and $4$ have the highest difference. Is there a word for that relationship, rather than just "highest difference" or "greatest range"?
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4answers
146 views

What is the difference between the relations $\in$ and $\subseteq$?

Don't they both mean that something is an element of a set? Are they interchangeable in some or all situations? Like: $x \in A$ ($X$ is an element of the set $A, X$ is in $A, A$ contains $X$) $x ...
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1answer
25 views

Which of these relations are maps?

List all relations $\{a,b\} \to \{c,d\}$, assuming $a \neq b$ and $c \neq d$. Which of them are maps? So I know the cartesian product gives $\{(a,c),(a,d),(b,c),(b,d)\}$. And the relations will be ...
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0answers
27 views

A question regarding rational functions being onto.

We are given the following function: $$K:\Bbb{R}\setminus \{2\}\mapsto \Bbb{R}\setminus \{1\}$$ defined by $K(x)=\dfrac{x+12}{x-2}$ Is the function K onto (surjective) or not?
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2answers
38 views

Describing equivalence classes of ad=bc mod n

Let $G = \{1,2,3,4\}$, and let $H = G\times G$. Define a relation $R$ on $H$ as follows: $$ (a,b)R(c,d) \text{ if and only if } ad \equiv bc \mod 5. $$ a. Show that $R$ is an equivalence relation. ...
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3answers
96 views

Does every subset start with an empty set? [closed]

Does every subset start with an empty set? If $A = \{a,b\}$, then the subsets of $A$ are $\{a\},\{b\},\{a,b\}$. Not sure if that is correct.
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1answer
22 views

Probability involving unions and intersections

Out of a group of people, 60% like Vegetables, 70% like Chocolate, and 40% like vegetables and chocolate. Find the probability that a randomly selected person in the group neither likes Vegetables nor ...
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1answer
55 views

Is this statement correct?

Define $N=\left\{\dfrac{1}{n}\mid n\in\mathbb{N}^+\right\}$, so $N\subset\mathbb{Q}$. We can also add to this $N\subset[0,1]$ or $N\subseteq(0,1]$ or $N\subset(0,1]$. Is saying $N\subseteq[0,1]$ also ...
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1answer
56 views

Is this a correct proof of the Cantor-Bernstein-Schröder theorem?

Sorry its a little messy. I haven't really cleaned it up yet. Theorem: if $f:A\to B$ and $g:B\to A$ are injections, then there is a bijection from A onto B. proof: Let $b \in B$. Since $g$ is ...
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1answer
29 views

Want a succinct way of defining this particular set

Suppose $S$ is defined as being the set: $\{$ $\quad \{ (x, 1), (y, 1), (z, 1) \},$ $\quad \{ (x, 1), (y, 1), (z, 2) \},$ $\quad \{ (x, 1), (y, 1), (z, 3) \},$ $\quad \{ (x, 1), (y, ...
1
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1answer
27 views

Bijection between well-ordered set and ordinal

We are working in ZF. Problem (part 1). Suppose we have well-ordered set $(M, <)$. How to show that there exists a bijection $f$ between this set and some ordinal $x$ that preserves order? I can ...
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3answers
280 views

A question about cardinal number.

Let $X$ be a infinite set and $n$ be a positive integer. We denote the cardinal number of $X$ by $|X|$ and denote the family of all subsets of X which contains n elements by $\mathfrak{F}$. Then ...
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2answers
53 views

A question on the proof of commutativity of the sum of natural numbers?

I made this question yesterday and today I've been thinking about another aspect of it. But this question is totally related to the previous one: I am trying to make a clarification about a proof of ...
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1answer
64 views

Prove that the set of all finite sequences of real numbers has the same cardinality as the set $\mathbb{R}$ of reals. [duplicate]

Prove that the set of all finite sequences of real numbers has the same cardinality as the set $\mathbb{R}$ of reals. I can not understand the purpose of the question.
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1answer
24 views

Complement of a set and inverse image.

I'm currently taking a real analysis class and we are working on measurable functions (the notes can be found here under "Measurable Functions"; the exercise is the first one on the last page). ...
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2answers
39 views

Constructing a one-to-one correspondence between closed interval and half open interval

Construct a one-to-one correspondence between the closed interval $ [0,1] $ and the half-open interval $ [0,1)$. Hint: Take $B=\{1\}$
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0answers
17 views

Proving union of two $\sigma$ algebras in $X$ is an algebra, then it is a $\sigma$-algebra.

I tried two countable union of $M_1, M_2 \in 2^X$ such that $A\in M_1, B\in M_2$ where $ A=\bigcup_{k=1}^{\infty}A_k, B=\bigcup_{k=1}^{\infty}B_k $. But Im stuck on that I should prove ...
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0answers
13 views

What is the set of the accumulation points of this set?

Let $A := \{ \frac{1}{n} \mid n \in \mathbb{N} \}.$ I wish to find the boundary of $A$. Since $\overline{A} = A \cup \{ 0 \},$ there remains to find $\overline {\mathbb{R}\setminus A}.$ Since ...
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1answer
42 views

Given the relation $\{(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\}$ determine whether it is reflexive, transitive, symmetric, or anti-symmetric.

Given the relation $\{(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\}\,$ determine whether it is reflexive, transitive, symmetric, or anti-symmetric. I found this set to be reflexive and symmetric. But not ...
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5answers
55 views

In set theory; How to show $|B_3 \cup (B_1\cap B_2)| = |B_1| + |B_2| +|B_3| - |B_1 \cup B_2|$?

I am unable to solve this problem. Can anyone show me how to prove this above formula?
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1answer
20 views

Is ordering of (possibly infinite) sets by cardinality a total ordering?

Given sets $A$ and $B$. Can you show that either there exists an injective map of $A$ into $B$ (that is, a map such that each element of $A$ maps to an element of $B$ and no two elements of $A$ map to ...
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0answers
20 views

Recommended Level of Depth on Set Theory for Applied Math (LA,Prob)

I hope this is appropriate to post on these specific tags. If one's objective is to become highly proficient in the application of Linear Algebra, Probability, Statistics and Optimization, how much ...
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1answer
16 views

Number of pairwise disjoint subsets of a set R, where the total cardinality of the pairwise subsets is less than or equal to some k < |R|

I'm not a mathematician and so I am not sure if my question is worded right. I am a software engineer who just started doing a PhD and so there is a lot of theoretical computer-science stuff that ...
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4answers
123 views

Are there two meanings to induction?

I've seen mathematical induction in two forms. First form: It seems that if $P(0)$ holds and $\displaystyle ...
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1answer
36 views

Good books of naive set theory

Is there a good book naive set theory which prove important theorems and propositions like: The rational numbers are countable The real numbers are not countable $card \ (0,1)=card\ ...
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1answer
35 views

Term for relationship of set overlap without containment?

What is the term for the relationship between two sets that share at least one common element, but neither set is a proper or improper subset of the other? $$ A?B=_{def}\exists x \exists y \exists z ...
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4answers
69 views

Show that all the intervals in $\mathbb{R}$ are uncountable

Question: Show that all the intervals in $\mathbb{R}$ are uncountable. I have already proven that $\mathbb{R}$ is uncountable by using the following: Suppose $\mathbb{R}$ is countable. Then ...
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1answer
24 views

$A$ and $B$ are infinite sets with $B \subseteq A$. Which of the following statements are true?

$A$ and $B$ are infinite sets with $B \subseteq A$. Which of the following statements are true? $A \sim B$ $A \sim A \setminus B$ If $B$ is countable then $A \sim B$ If $A$ is ...
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1answer
40 views

Is it possible to extend well ordering principle/induction to all well ordered sets?

Today I was thinking about well ordering of naturals,and how by induction we can prove some properties of natural numbers.Now I started wondering if this is property of natural numbers,which are well ...
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3answers
82 views

Teaching cardinality

I would like to give a class of 60 minutes to my undergraduate students about cardinality. I would like to begin with the definition of cardinality and end with one or two good application of this ...
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1answer
20 views

A Criterion For a Set To Have all the Atoms of a Boolean Algebra

Let $\Omega$ be any set and let $\mathcal A$ be an algebra of sets in $\Omega$. An element $E\in \mathcal A$ is said to be an atom if there is no non-empty element $A\in \mathcal A$ such that ...
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1answer
35 views

How to solve unitary work problem by Set theory?

How to solve by set theory - if $A,B$ and $C$ can complete a work individually in $4, 5$ and $6$ days, how many days they will take together to finish the work?
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2answers
21 views

If $A_n \downarrow A.$ then $A_1 - An \uparrow A_1 - A$? Set theory.

Let $A_1, A_2 , \dots$ be subsets of a set $\Omega$. If $ A_1 \subset A_2 \subset \dots$ and $\bigcup_{n = 1}^{\infty} A_n = A $ then we write $A_n \uparrow A.$ $ A_1 \supset A_2 \supset \dots$ and ...
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1answer
22 views

Show that the built set $\{ (x, y) \in \mathbb{R}^{2} \mid x^{2} + xy + y^{2} = 1 \}$ is compact in $\mathbb{R}^{2}$

I see that the set $\{ x^{2} + xy + y^{2} = 1 \}$ is bounded, because given any $y$ there are exactly two $x$ such that $x^{2} + xy + y^{2} = 1$ and so is the case where $x$ is given. But I do not ...