4
votes
3answers
390 views

Definition of set.

A set is defined as a collection of distinct objects. Why have we defined a set to contain only distinct objects? Why is a collection of objects which may have identical objects not called a set? ...
0
votes
4answers
83 views

How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
0
votes
2answers
30 views

Confused about class definition

I find this in my Set theory material: [0] = {x:0==x(mod2)} = {x:2|0-x} , where I'm replacing equivalence sign ("=" with extra horizontal line) with double ...
3
votes
2answers
48 views

Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, ...
0
votes
0answers
18 views

Depth in acyclic graphs

I am struggling to understand a definition in a paper: Given a acyclic (directed!) graph $D=(V,E)$ we define a sequence $Q_i \subset V(D)$ of sets: $$Q_0 = \emptyset,$$ $$ Q_i \textrm{ is ...
1
vote
1answer
67 views

Is this a correct definition of the natural numbers in ZF?

Set $s$ is a natural number if $s$ is transitive and for every $x$, $y$ and $z$ $y\in{s}\rightarrow(y$ is transitive$)$, and if $x\in{P}s\wedge(x$ is transitive$)\wedge{z}\in{P}x\wedge(z$ is ...
0
votes
3answers
42 views

What is the usual definition of “ordinal number”?

My definition for the ordinal number is "von-Neumann ordinal". I thought this is the only definition for the ordinal, but i found some other definitions in wikipedia. What is the usual definition of ...
1
vote
2answers
37 views

Point as an element of an affine space vs point as an element of a topological space?

I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong ...
2
votes
3answers
57 views

I'm confused about the definition of poset.

The definition of poset : $\qquad$A set with a partial ordering. Partial ordering is a binary relation $\preceq$ over a set ($P$). If I understood the definition of relation correctly, then ...
1
vote
1answer
55 views

Example of a set that is Dedekind-finite but not Tarski-finite?

Can you give an example of a set that is Dedekind-finite, but not Tarski-finite?
5
votes
1answer
79 views

An alternative definition of finite?

Does the following definition adequately characterize the notion of finite? Is it equivalent to, say, Dedekind-finiteness? A set $S$ is finite if and only if for all $x_0\in S$ and all $f:S\to S$, if ...
-3
votes
1answer
78 views

Alternative definitions of functions requiring non-empty domains?

It is easy enough to prove in set theory, but it seems counter-intuitive to me that an empty set could be the domain of a function. Is there any literature requiring that functions have non-empty ...
2
votes
1answer
42 views

What is the defenition of $\mathcal{c}$ and $\aleph_1$ if we assume ZFC without CH.

I am reading an intro to a chapter of Andreas Blass called "Combinatorial Cardinal Characteristics of the Continuum" and I am getting a bit confused. When I studied "Discrete Mathematics", it was ...
0
votes
1answer
87 views

How to tell if relation on set is a partial order when relation is defined as a set of ordered pairs?

Determine whether $R$ is a partial order on set the $S$ and justify your answer. $S=\{1,2,3\}$ and $R=\{(1,1),(2,3),(1,3)\}$. So the job is to consider if $R$ is reflexive, antisymmetric and ...
0
votes
1answer
42 views

Forward and backward composition in relational algebra

http://imgur.com/lMbx4Q5 I am having trouble understanding what forward composition and backward composition mean. The picture above is from my unit notes and I just fail to see any intuition reading ...
0
votes
2answers
35 views

What is the intersection between the set of all expressions, of all equations and of all functions?

I am studying the definition of mathematical expression, of equation and of function and I want to draw a venn diagram with the intersection between the set of these objects. Some people say every ...
1
vote
2answers
106 views

Meaning of $\{ a,b \}$, and comparison with $(a,b)$

What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
0
votes
0answers
49 views

Formal and general definition of natural domain (natural set) of a function

Can anyone give me a (as much as possible) formal and general definition of natural domain of a function? Let's say that a function is a triplet $(X,Y,f)$ where $f \subseteq X \times Y$ such that ...
1
vote
2answers
51 views

“Preimage” of a binary relation

Consider the binary relation $R \subseteq X \times Y$. Is there a standard name and notation for the set $X' = \{x\ |\ (x, y) \in R\}$? ProofWiki calls $X'$ the preimage of $R$, denoted as ...
4
votes
1answer
210 views

Set-builder notation function definition

I know that a function is a subset $f \subseteq X \times Y$ such that \begin{eqnarray} \forall x \in X, \exists ! y \in Y | (x,y) \in f \end{eqnarray} First, is it possible to express what a ...
1
vote
1answer
37 views

What is the lower bound of the subset $2^n,\; n\in\mathbb{N}$

Let: $$ A = \{2^n,\; n\in\mathbb{N}\},\quad A\subset \mathbb{R} $$ Is the lower bound: $(-\infty,0]$ $(-\infty,1]$ $(-\infty,1)$ ? I think it can be the first because ...
2
votes
1answer
54 views

Differing definitions for 'Algebra of subsets'

For a collection, $A$ of subsets of a set $X$ to be an algebra of subsets it must satisfy the following properties: $A$ is non-empty If $E \in A \implies E^c \in A$ If $E, F \in A \implies E \cup F ...
2
votes
2answers
100 views

How do the terms “countable” and “uncountable” not assume the continuum hypothesis?

Every countable set has cardinality $\aleph_0$. The next larger cardinality is $\aleph_1$. Every uncountable set has cardinality $\geq 2^{\aleph_0}$ Now, an infinite set can only be countable or ...
5
votes
1answer
157 views

Difference between a type and a set

I've been trying to understand this distinction for a while, buts its still not making sense to me. Originally, I thought the distinction between type and set was as follows. The relationship ...
0
votes
0answers
47 views

About definition of inductive set (with sets or ur-elements)!!

---let $A$ a set, $A$ is inductive if $\emptyset \in A$ and $\forall x \in A (x^+ \in A) $ ---let $A$ a set, $A$ is inductive if $\emptyset \in A$ and $\forall Y \in A (Y^+ \in A) $ an example of ...
1
vote
1answer
224 views

Formal definition for indexed family of sets

Essentially I'd like to know the formal definition of the object $\{A_{i}|i\in I\}$ .This is my context: 1.- From Wikipedia (Here) I understand that a family of elements in $S$ indexed by $I$ and ...
0
votes
0answers
71 views

$\mathbb{R}$ as set of Dedekind cuts on $\mathbb{Q}$

-- "let $A,B \in \mathbb{Q}$, $A < B$ if $A\leq B$ and $A \neq B$ -- "let $\preceq$ be a ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq $ if $\forall a ...
2
votes
2answers
209 views

What are authoritative publications regarding foundational mathematics?

I have a computer science background. In our world, there usually is an organization publishing standard documents for certain areas (e.g. W3C has Web standards, IETF publishes Internet-related ...
1
vote
1answer
53 views

Is it true that the definition of an open subset in a metric space is different from the combination of the definitions of subsets and opens sets?

Dear reader of this post, I have a question concerning the equivalence of two definitions of open subsets (in metric spaces). To avoid confusion, I will state the two definitions and then ask my ...
1
vote
1answer
51 views

What is the difference between a reflexive relation and an identitive relation

Given a set $X$ and a relation $R$ over $X$, we say that $R$ is reflexive if \begin{equation} xRx\ \forall\ x\in X. \end{equation} What does 'identitive' mean? Is it the same as antisymmetry? Seen ...
2
votes
2answers
149 views

How to extend definition of n-tuple to the case $n=0$?

The classical definition of n-tuple $(x_i)_{i < n}$ starts at $n=2$. In this case $$(x_0,x_1) := \{\{x_0\},\{x_0,x_1\}\}$$(1). For $2<n=k+1$, $(x_i)_{i < n}:=((x_i)_{i < ...
4
votes
4answers
3k views

Is the empty set a subset of itself?

Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty ...
1
vote
5answers
574 views

Definition Of Symmetric Difference

The definition of a symmetric difference of two sets, that my book provides, is: Set containing those elements in either $A$or $B$, but not in both $A$ and $B$. So, in set builder notation, I figured ...
2
votes
1answer
704 views

Definition of the complement of a set

My book defines the complement of a set as, "Let $U$ be the universal set. The complement of the set $A$, denoted by $\bar{A}$, is the complement of $A$ with respect to $U$. Therefore, the complement ...
0
votes
2answers
52 views

Question about power of sets

If two sets are finite and they have the same power, can we say that the two sets are equivalent? Is every finite set countable?
10
votes
8answers
1k views

Why do we accept Kuratowski's definition of ordered pairs?

I've been struggling understanding Kuratowski's definition of ordered pairs. I understand what it means but I don't see why I should accept it. I've seen this question and this one, most importantly ...
0
votes
2answers
202 views

About the definition of n-tuple

I've read from the theory of sets that the definition of ordered pair is foundational. To define formally the term of $n$-tuple I suppose we need to use the concept of ordered pair as well as the ...
2
votes
1answer
81 views

A question on linear ordered space

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$. And we know every space contains a dense left-separated subspace. My question ...
2
votes
3answers
172 views

What is the definition of first/last element in a poset?

I have read the terms first element/last elements in the context of a basic course in set theory. When I learned about posets I didn't encounter those terms. I tried looking up the definitions but I ...
3
votes
2answers
752 views

Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
1
vote
4answers
663 views

what is total order - explanation please

sorry for the dumbest question ever, but i want to understand total order in an intuitive way, this is the defition of total order: i) If $a ≤ b$ and $b ≤ a$ then $a = b$ (antisymmetry); ii) If $a ...
1
vote
4answers
760 views

Partition of a set, definition not clear

From wikipedia: Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and: The union of the elements of P is equal to X. (The elements of P are said ...
5
votes
2answers
187 views

I need to disprove an alternate definition of an ordered pair. Why is $\langle a,b\rangle = \{a,\{b\}\}$ incorrect?

So we know that the an ordered pair $(a,b) = (c,d)$ if and only if $a = c$ and $b = d$. And we know the Kuratowski definition of an ordered pair is: $(a,b) = \{\{a\},\{a,b\}\}$ ...
4
votes
2answers
173 views

Which algebraic structure captures the ordinal arithmetic?

Consider the set class $\mathrm{Ord}$ of all (finite and infinite) ordinal numbers, equipped with ordinal arithmetic operations: addition, multiplication, and exponentiation. It is closed under these ...
3
votes
1answer
215 views

Examples of preorders in which meets and joins do not exist

Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
23
votes
3answers
3k views

difference between class, set , family and collection

In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
5
votes
1answer
188 views

What set operation is this?

Given two sets $ A = \{\{1\} , \{2 , 6\} \}$ and $ B = \{\{2\} , \{3\} , \{4 , 5\} \}$, what set operation can produce $$ C = \{ \{ 1 , 2 \} , \{ 1 , 3 \} , \{ 1 , 4 , 5 \} , \{ 2 , 6 , 2 \} , \{ 2 , ...
-1
votes
2answers
200 views

Definition of a point and object

Is there any theory in which a point has a definition? What is the definition of "object" as seen in category theory?
6
votes
2answers
179 views

What is $\mathbb{N}^{<\mathbb{N}}$?

What is the definition of this symbol $\mathbb{N}^{<\mathbb{N}}$? How is it related to the infinite product $\mathbb{N}^{\mathbb{N}}$?
0
votes
1answer
253 views

Why is this definition of an additive inverse significant

In the process of learning Real Analysis, I encountered a definition of an additive inverse of a cut $\alpha$ to be $$\text {add inv of } \alpha \colon= \{p:\exists r>0 \text{ s.t.} (-p-r)\notin ...