# Questions tagged [relations]

For questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric, etc), a composition of relations, or anything else concerning a relation on a set.

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### Can a partially ordered set contain an infinite cycle?

A partially ordered set is defined as a set with a relation that is symmetric, transitive, and anti-reflexive. The transitivity and anti-reflexivity rule out cycles. We can't have "a < b < ...
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### What kind of operation is cube root extraction?

I came across this question in a random test and the correct answer was marked as "Binary Operation". I am pretty sure that to find the cube root of a number you only need that number alone ...
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### Completeness of Binary Relations

Completeness of binary relations often is defined as: The binary relation R of a set A is complete iff for any pair x,y ∈ A: xRy or yRx. My question is: what does one mean by „pair“? To me it seems ...
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### Assume $g: A \to B$ and $f: B \to C$. If $f\circ g$ is surjective, then $f$ would be injective. True or false?

Assume $g: A \to B$ and $f: B \to C$. If $f\circ g$ is surjective, then $f$ would be injective. Would this proposition be true or false?
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### Question about Equivalence class [closed]

I believe I understand the concept of equivalence class. However, I found this example from my lecture slides very confusing. Could someone explain to me why ...
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### Proving reflexivity, symmetry, and transitivity for the relation $\sim$ on $\Bbb{R}$ such that $x\sim y$ iff $x+y\in\Bbb{Q}$

I am going through past papers for my university exam, and a question in this format appears often: Define a relation $\sim$ on $\Bbb{R}$ by $x\sim y$ if and only if $x+y \in \Bbb{Q}$. Justify your ...
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### Is there any proof of $\#(F/N)=2n$ which doesn't use any group other than $F/N$ itself? (Michael Artin "Algebra 1st Edition")

I am reading "Algebra 1st Edition" by Michael Artin. The following proposition is Proposition (8.3) on p.221 in this book. (8.3) Proposition. The elements $x^n,y^2,xyxy$ form a set of ...
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### Question about a proposition about free groups, generators and relations. Is it true or false that $N=\ker\phi$ holds? Michael Artin "Algebra 1st Ed."

I am reading "Algebra 1st Edition" by Michael Artin. I feel free groups, generators and relations are very difficult. The following proposition is Proposition (8.3) on p.221 in this book. (...
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### How do I prove symmetry without a defined set?

I have a formula: ∀x,y, z(xRy ∧ xRz → yRz) If the formula holds for a relation, then the relation is Euclidean. If a relation is Euclidean and reflexive, what are the steps for proving it is also ...
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### What does the author want to say about generators and relations in group theory? ("Algebra 1st Edition" by Michael Artin) [closed]

I am reading "Algebra 1st Edition" by Michael Artin. I want to know about generators and relations because I think I need to know about generators and relations when I use the GAP software. ...
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