4
votes
2answers
513 views

'Does not necessarily equal' symbol

What symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \in ...
1
vote
2answers
129 views

Notation of a function that maps a random element

Let there be a functions $f$ and $g$ such that, $$f:A \times B \mapsto \Re$$ $$g: B \mapsto A$$ where $\forall b \in B$, $g(b)$ is some $a$ such that, $\forall a' \in A, f(a,b) \geq f(a',b)$. (This ...
0
votes
0answers
26 views

“Anti-cumulative” Relation Image using Intersection

Given a binary relation $R \subseteq X \times Y$, the familiar image of some $A \subseteq X$ is defined as $R[A] = \{y\ |\ (x, y) \in R, x \in A\}$. Naturally we have the property $R[A] = \bigcup_{x ...
0
votes
0answers
41 views

Embedding vs restriction

Embedding is the morphism $( A ; B ; \operatorname{id}_A)$ of the category $\mathbf{Rel}$ for sets $A \subseteq B$. I call restriction the morphism $( A ; B ; \operatorname{id}_B)$ for sets $A ...
2
votes
1answer
47 views

Can objects repeat in commutative diagrams?

Are objects allowed to repeat in commutative diagrams? This seems to be necessary when representing endomorphisms such as the morphism $f : X \to X$ in the category $\mathbf{Set}$, such as when $f$ is ...
0
votes
0answers
24 views

Restricting binary relations by composing with an “inclusion binary relation”

If $X' \subseteq X$ then we may define an inclusion map $\iota : X' \to X$ where $\iota(x) = x$. One use of $\iota$ is that we can express the restriction of some $f : X \to Y$ to $X'$ as $f|_{X'} = f ...
1
vote
2answers
56 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 ...
0
votes
1answer
119 views

Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
6
votes
2answers
195 views

Notation for a relation

I'm reading up on "Set Theory and Logic" by Stoll and came upon notation for relations that I haven't seen before. I've seen $x\sim{y},$ and $xRy$ before but Stoll uses this one. $$(x,y)\in{\rho}$$ ...
2
votes
2answers
85 views

How to represent objects formally? [closed]

Computer indexed arrays can be represented formally using vector notation. x= [1,3,2] $x=(1,3,2)$ How can I show an object/associative array mathematically? y=['a'=>5,'b'=>4] ~ ?
0
votes
2answers
124 views

What is this problema asking for? I don't understand the question. (set notation, composite relations)

In the following problem, what does the "circle" between set names represent? What exactly is this problem asking me to do? Consider the following relations on Z: $R1 = \{(x, y) | y = x + 1\}$ ...
0
votes
2answers
150 views

Elementary Set Theory - Relations

I'm not exactly sure what to search for this problem I'm having, as I don't know the keywords, so I figured the best action would be to ask a question. I have this question: ...
1
vote
2answers
224 views

How many total order relations on a set $A$?

Let's define a set $T_A$ which is the set of all total order relations on $A$. This set is a subset of the set of all $2$-adic relations on $A$: $$T_A \subset \mathcal P(A^2) $$ 1-Which is the ...
6
votes
3answers
132 views

Abuse of notation in declaring a variable is a function of another?

The standard way to write $ \text{ y is a function of x} $ is $ y = f(x) $ This is taken to mean that $y$ is the value of function $f$ evaluated at $x$. For simplicity let's take $f$ to be some ...
3
votes
2answers
225 views

What is the correct notation for a multivariable function?

Many mathematical texts define a multivariable function $f$ in the following way $$f := f(x,y)$$ However, if we focus on the fact that a function is really a binary relation on two sets, (say the ...
3
votes
1answer
155 views

Diagrammatic (Postfix) Composition of Functions

Consider the functions $f : X \to Y$ and $g : Y \to Z$. According to the Wikipedia articles on Function Composition, the application of $f$ to an input $x$ can be written as $xf$ (as opposed to the ...
0
votes
5answers
177 views

What is the difference between set membership symbol $\in$ and $R$?

I know that $(a,b)\in R$ means an ordered pair of elements $a$ and $b$ belonging to the set $R$ but sometimes I see some expression like $a R b$ ? What does this notation/expression mean ? How to ...
1
vote
2answers
284 views

What's the equivalent of the adjacency relation for a directed graph?

I've found several sources describing a relation notated $\sim$ signifying adjacency in an undirected graph, but nothing explicitly describing an equivalent for a directed graph. I've been using ...
1
vote
1answer
151 views

Name of binary relation: if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$

Is there a term for a binary relation $R\subset A^2$ on some set $A$ such that if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$ ? Are there any examples of it? Are there any related ...
0
votes
1answer
109 views

Need Help in Tabular Notation to represent a Relation

I am stuck with a question. It is stated below. Let $R$ be a relation defined on the set of integers $\mathbb Z$ by the rule $aRb \iff |a-b|\leq 2$. Write the relation $R$ as a set. Now there ...