# Tagged Questions

This tag is intended for questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric,...), composition of relations and similar stuff. More-or-less the things about relations taught in the first elementary set theory or discrete math course.

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### Venn diagram for a relation

My high school math book says the following diagram is a Venn diagram. But I think this is not correct. Is it right? If not, what is the following diagram that represents a relationship called?
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### Steps to determine if a relation of a set is reflexive,symmetric or transitive?

I am having problem understanding these concepts. For example, let $A = \{2,3,4,5,6,7,8\}$. The definition I found says that $x R y \iff 3 | (x-y)$. How do I know if the relation $R$ on $A$ is ...
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### Properties of a relation

$\cong\;=\{((x_1,y_1), (x_2,y_2))\in \mathbb R^2 ×\mathbb R^2 |x_1^2-x_2^2=3y_1^2-3y_2^2\}$ finitary relation meaning $(x_1,y_1) \cong (x_2,y_2)$ if $x_1^2-x_2^2=3y_1^2-3y_2^2$ Is this finitary ...
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### equivalence relations proof over the same set

I want to proof the following theorem: Let R be an equivalence relation on set A. Then {R[a]:a that belongs to A} is a partition of A. So long I have manage to proof that each a that belongs to A, ...
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### Recursive definition of the relation greater than on N X N

Give a recursive definition of the relation greater than on N X N using the successor operators s? I started this question throw this way: basis: (1,0) ∈ N x N could someone help me in recursive ...
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### are these binary relations?

I have found the following examples of Binary Relations, but I am not pretty sure is the conclusion the author arrived is correct. X is a number of people x N y, implies that x lives next to y; for ...
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### Symbol for unknown relation?

When solving equations like \begin{align} 4x-4 &=\frac{(2x)^2}{x} \\ -4 &= \frac{4x^2}{x} -4x \\ -4 &= 4x -4x \\[0.2em] -4 &= 0\end{align} using the equality-symbol feels like ...
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### Difference between Inclusion and continuation

Halmos defines the order continuation as follows: We shall say that a well ordered set A is a continuation of well ordered set B if B is a subset of A, if, in fact, B is an intial segment of A and ...
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### $aRb$ iff $a=b(10)^k$ for some $k \in \mathbb{Z}$ [Prove Equivalence Relation]

The question: $R$ is a relation on $\mathbb{N}$ defined by $aRb$ iff $a=b(10)^k$ for some $k \in \mathbb{Z}$. Prove that $R$ is an Equivalence relation. The problem: I can define an ...
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### How to find relation between 2 numbers

I have been practicing programming for many months now and what I found difficult is not about solving problem. But it is how to find the "how to solve problem" to make computer solves that for me! ...
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### Does an asymmetric relation entail an antisymmetric relation?

So if there exists an asymmetric relation within a set, does it also entail that there will be an antisymmetric relation in that same set? If so, then it is possible to find out whether a set ...
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### Show that $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : 4 \mid(5x+3y)\}$ is an equivalence relation.

Let $R$ be a relation on $\mathbb{Z}$ defined by $$R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : 4 \mid (5x+3y)\}.$$ Show that $R$ is an equivalence relation. I'm having a bit of trouble with ...
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### Determining if the relation is an equivalence one.

Determine if the relation : $$x \sim y \iff |y-x| \text{ is an integer multiple of } 3$$ is an equivalence one. Now, I think this is an equivalence relation but I am having troubles formally ...
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### Why is this relation a function?

I need to determine whether or not the relation $\{ (a^2,a) | a \in \Bbb {R}, a \geq 0\}$ is a function from $\Bbb {R}$ to $\Bbb {R}$. I think that it is a function. But I don't know how to justify ...
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### Is the relation a function

I'm trying to determine if the relation $\{(\frac{a}{b}, a-b) | a,b \in \Bbb {Z}, b \neq 0\}$ is a function from $\Bbb {Q}$ to $\Bbb {Z}$. I know that a relation is a function from A to B if dom(f)=A ...
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### How many functions are transitive?

Let the set of all functions defined as: $\left\{a,b,c,d\right\} \rightarrow \{a,b,c,d\}$ How many functions are transitive? I've been told to use the fact that a function is transitive iff "it's ...
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### Prove that the binary relation “is a subset of” is a…

Prove that the binary relation "is a subset of" is a partial order (POSET)? Should I try to prove this in reference to the power set of a general set? When is this relation a total order?
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### An accessible example of a preorder that is neither symmetric nor antisymmetric

For a project I am working on, I need an example of a preorder (reflexive and transitive relation) that is neither symmetric (like an equivalence relation) nor antisymmetric (like less than or equal ...
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### Example of relation that is neither transitive nor intransitive?

I have been struggling to think of an example of a relation that is neither transitive nor intransitive, does anyone have any tips? I ended up finding one website that described this as non transitive,...
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### Is it Preference relation?

I need to check if the relation $\succeq ( \space \succeq \space \subset X × X, \space X=VB[0,1] )$ define as below $$f \succeq g \Longleftrightarrow Var(f+g) \geq Varf$$ is preference relation....
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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 $a, b$?
### Quotient set cardinal in $\mathbb{Z}_{12}$
In $\mathbb{Z}_{12}$ define the equivalence relation xRy if $x^2 = y^2$ Then what is the cardinal of the quotient set?