# Tagged Questions

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### “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 ...
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### 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 ...
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### 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. $$(x,y)\in{\rho}$$ ...
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### 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] ~ ?
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### 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: $R1 = \{(x, y) | y = x + 1\}$ ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...
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 ...