# Tagged Questions

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### What is the name of this property of relation?

What is the name of property of a binary relation $R$ that states that $\lnot(a\mathrel{R} b) \iff \lnot(b \mathrel{R} a)$ for all $a, b$?
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### Is there a name for sum over one set divided by the cardinality of another set?

What is the summation of one set real numbers divided by the cardinality of another set called? $$A \subset\mathbb R$$ $$\frac{\sum A}{|B|}$$ I will try and be specific to my problem because I lack ...
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### What does it mean for a function to be uniquely determined by another function?

In munkres topology, I went through an exercise which asks me to show that a function is uniquely determined by another function. I wonder, What does this mean? I googled it but No answer! Here is ...
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### Question about definition of binary relation

Wikipedia says: Set Theory begins with a fundamental binary relation between and object $o$ and a set $A$. If $o$ is a member of $A$, write $o \in A$. I thought that a binary relation is a ...
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### What if union of disjoint sets results in universal set?

I have a question related to set theory. If $A_1,A_2,A_3\dots, A_n$ belongs to universal set $U$, and if all of the sets are disjoint i.e. $A_i \cap A_j = \emptyset$ for all $i$ and $j$. And If ...
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### Initial Segments and Initial Sections of Posets

For a set A with a partially ordering <=, define the following 1) A subset s(x) of A = {y in A such that y <=x} 2) A subset S of A with the property that for every x in S then all y in A ...
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### Correspondence as a graph of a multifunction

Suppose I'd like to say that a projection of $R\subset X\times Y$ on $X$ is the whole $X$. That is, $R$ is a graph of a certain multifunction, or equivalently it is a left-total relation. I do ...
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### Identity relation of many variables

The identity relation on a set $A$ is $\operatorname{id}_A = \{(x;x) \,|\, x\in A\}$. This can be generalized for any (possibly infinite) index set $N$ as $\{(\lambda i\in N: x) \,|\, x\in A\}$ (here ...
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### Partial order up to equivalence

In certain contexts one runs into something like a partial order, but the antisymmetry property is weakened as follows: if $x \preceq y$ and $y \preceq x$ then $x \simeq y$, where $\simeq$ is a given ...
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### into function vs injective function

In many mathematical books that I have read and from lectures from professors, the words 'into' and 'injective' were used interchangeably, but in Patrick Suppes book Axiomatic Set Theory he gives a ...
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### Basic Cartesian prodcuts

I am having some issues grasping basic ideas of Cartesian products. I am reading a PDF my professor gave us explain sets/Cartesian products. If $\mathbb{R}\times \mathbb{R}$ can be written as ...
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### The Empty Relation?

In elementary set theory, a relation on sets $A,B$ is usually defined as a subset of $A\times B$. We know that there are $2^{|A\times B|}$ subsets of $|A\times B|$. One of these subsets is the empty ...
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### What is the name of this set?

What is the standard name for the set of all n-ary functions, where n is a natural number,of some set S, say the reals or the complexes? We have the notation S^S, but that is only the set of 1-ary ...
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### What is a set of overlapping sets?

If I have a set $X$ and a set $Y$ and $\forall y \in Y : y \subseteq X \land \exists y_1, y_2 \in Y : y_1 \ne y_2 \land y_1 \cap y_2 \ne \{\}$, what is the relationship between $X$ and $Y$ called?
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### Name of a set of the form {x,y}

I know that a doubleton is a set with exactly two elements, but what is the name of a set with either exactly 1 element or exactly 2 elements? In other words, what is the name of a set of the form ...
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### Given a subcollection of a powerset, do these “separation” relations have names?

Let $X$ denote a set and $\mathcal{F}$ denote a subcollection of $\mathcal{P}(X).$ Do the following relations on $\mathcal{P}(X)$ have a name? For $A,B \subseteq X$, call $A$ partially separated from ...
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### Problem in Set Theory determining elements [closed]

"H does not include D" in this statement is D is an element or a set? if it is a set what is the set notation?
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### Reflexivity: How can something be related to itself?

Background: I'm a philosophy student. I'm comfortable with math, but don't have much of a background in it. One of the topics I'm writing about (I-relation in theories of identity) closely mirrors ...
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### “Set” vs “collection” terminology: what is the difference?

Can someone tell me what is the difference in saying $A$ is a set of even numbers and $X$ is a collection of even numbers ?
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### Analogue of the term 'summand' for unions and intersections.

If we have a sum $\sum\limits_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the ...
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### “Collection”: What does it mean?

I've seen a lot of question of same ilk as the request I'm about to pose, but what I'd like to know is what does "any collection" mean in the following request: Prove that the intersection of any ...
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### The counted is to the countable as the ??? is to the (order)-isomorphic.

We sometimes need to distinguish the counted from the countable. A counted set is a set equipped with a particular bijection into (some of) the natural numbers; a set is countable if there exists such ...
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### Name for Cartesian Product variant that does not return an empty set if one of the sets is empty

I am looking for the name of this mathematical operation that behaves very similar to Cartesian Product. Given: A = {1,2} ...
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### Name for $X^\infty=\bigcup\limits_{k=0}^\infty X^k$

I'm making structures associated with groups, rings and so on in OCaml and in order to do so I started by defining sets and a few operations (intersection, union, difference, carthesian product, ...
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### The union of a countable set of countable sets?

Let $A$ be an countable set, and let $B_n$ be the set of all $n$-tuples $\left(a_1,\ldots,a_n\right)$ $B_n$ is the union of a countable set of countable sets. This question maybe about the ...
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### Formal notation when using the axiom of specification

The axiom of specification states formally that for every property $\varphi$ holds $\forall X\exists Y\forall x(x\in Y\longleftrightarrow x\in X\wedge\varphi(x))$. Since from the axiom of ...
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### Do these union- and intersection-like operations have a name?

I have two sets of pairs, e.g: $$A = \{ (a, 1), (b, 2), (c, 3) \}$$ $$B = \{ (b, 12), (c, 13), (d, 14) \}$$ I also have two operators which match the first element of pairs and return a pair of ...
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### Suppose A is a set of (x, y)', what is the name of the set that consists of all x in A?

Let $A$ be a set of a vector $(\mathbf{x}',\,\mathbf{y}')$. Here $\mathbf{x}'$ and $\mathbf{y}'$ could both be vectors. Is there a particular terminology for the set of all $\mathbf{x}'$ in the set ...
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### “$f$ is a function from $A$ to $B$” vs. “$f$is a function from $A$ into $B$”?

When we say that $f$ is a function from $A$ to $B$ is this different from saying $f$ is a function from $A$ into $B$ I know what injective ("1-1"), surjective ("onto"), and bijective ...
Let $S$ be a set. Let $\sim$ be a binary relation on $S$. Suppose $\sim$ follows these three rules. $x\sim x$ for all $x\in S$ (reflexivity). If $x\sim y$, then $y\sim x$ for all $x, y \in S$ ...