# 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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### I am confused about this notation for matrix representation.

I am confused about this notation from the image below. How can you represent something like (1,2),1 as a 2D matrix? For S1 = (0,1,2) and S2 = (0,1) I would expect two matrices like: [(1,0),(1,0),...
0answers
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2answers
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### Section and segment of a relation $R$

$\mathbf{Definitions:}$ $Z$ is a $R$-section of a set $X$ iff $Z\subseteq X$ and $x\in Z$ whenever $x\in X \ \land y\in Z \land \ xRy$, $\$for some relation $R$ on $X$ The $R$-segment of a set $X$...
0answers
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1answer
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### The difference between congruence and equivalence class?

I've got an excercise solved by my teacher, it says I've got to prove a relation $R$ of elements in $\mathbb{R}^2$ is a congruence. In the solved exercise he just proved Reflexivity, transitivity and ...
3answers
55 views

### Can Relations have a Domain and Codomain?

Does it make sense to talk about the domain and codomain of a relation? For example if the relation, $R$ $$x^2+y^2=r^2$$ only takes values $(x,y)\in\mathbb{R}^2$ its domain would be $\mathbb{R}^2$ ...
0answers
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1answer
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### How to construct a symmetric, transitive and reflexive relation

If R is a binary relation in a set X ≠∅ that is symmetric, transitive, then R is reflexive. This is false and I have to change the argument to make it true. How can I do this? Thanks!
2answers
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2answers
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### What is the period of $f(x) = \cos (x) \cos(2x) \cos(3x)$? [closed]

What is the period of $f(x) = \cos(x) \cos(2x) \cos(3x)$? Please tell me the method plus the logic behind solving these kind of problems .. Plus is there any property for even functions like even ...
2answers
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4answers
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### Definition of smallest equivalence relation

I came across the term 'smallest equivalence relation' in the course of a proof I was working on. I have never thought about ordering relations. I googled the term and checked stackexchange and couldn'...
0answers
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### Let $L= \{(x,y)\in \mathbb R \times \mathbb R : x \leq y\}$, and let $C=\{x \in \mathbb R : x > 7 \}$. Prove that C has no L-smallest.

Definition: Suppose R is a partial order on a set A, $B \subseteq A$ and $b \in B$ . Then b is called an R-smallest element of B iff $\forall x \in B [b R x]$ Goal is to prove the following: Let ...
1answer
25 views

### Discrete Maths Relations on the set {1,2,3,4}

I just want to make sure that I am doing these correctly. Here is what I have: Reflexive, symmetric, antisymmetric and transitive: And i have - {(1,1) (2,2) (3,3) (4,4)}. not Reflexive, not ...
0answers
19 views

### What is a transitive relation on set S

MY answer: Given r,s,t$\in S$ a transitive relation on the set $S$ is when the elements $rRs$ and $sRt$ then $rRt$ i.e., $rRs\land sRt\rightarrow rRt$ Does my definition words correct?
0answers
22 views

### Define symmetric relation R on set S

Answer: A symmetric relation R on S is that For all $x,y\in S$ such that $xRy$ implies $yRx$. meaning if the element x is related to y, then it is also true the other way around that element y is ...
1answer
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### Can someone explain why is $R=\{(1,1), (1,2)\}$ transitive?

Using a digraph I understand transitive relation to be a loop, but $R=\{(1,1), (1,2)\}$ is not a loop. Thank you for your time!
0answers
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### Restriction to equivalence relation is equivalence relation

Let $\mathcal{R}$ be relation on $A$ and $A_0 \subseteq A$. The $\mathbf{restriction}$ of $\mathcal{R}$ to $A_0$ is defined to be the relation $\mathcal{R} \cap (A_0 \times A_0)$. \$\mathbf{...
1answer
39 views

### Can someone explain antisymmetric versus symmetric relation of sets?

If $$A = \{1,2,3,4\}$$ and $$R = \{(3,3), (4,4), (1,4)\}$$ This example is antisymmetric but not symmetric. However, the definition of Antisymmetric taken from Merriam-Webster is this: ...
2answers
24 views

### Hasse diagrams for sorts of size 3 and 4?

Does anyone know what does the following mean? I understand that a Hasse diagram represents a given partial order but I don't seem to get this example. Below is a Hasse diagrams for sorts of size 3 ...
2answers
23 views

### What is the reflexive closure of the empty relation ∅ over a set A?

What is the reflexive closure of the empty relation ∅ over a set A? I understand that R is reflexive if A=∅, and isn't if A is nonempty. But what about the reflexive closure of R?