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Question: Let ∗ be a binary operation on a set A. Assume that ∗ is associative with identity. Let R be the relation on A defined on elements a,b ∈ R as follows: aRb if there exists an invertible element c ∈ A such that c∗b = a∗c. Prove that R is an equivalence relation. What happens if c is not required to be invertible?

I've figured out the first part of the problem, asking to prove that the relation is an equivalence relation. I'm confused as to how the answer will change if 'c' is not necessarily invertible. I believe that the relation will still be reflexive, but any guidance on transitivity and symmetry will be appreciated.

Thanks in advance.

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If for $\forall c\in A$ $c$ isn't invertible about $R$, $R=\emptyset$ by definition. –  Aran Komatsuzaki May 28 at 2:42
    
@AranKomatsuzaki that is not true... by hypothesis, at least the identity is in the set, and the identity is related to itself... e*e=e, so choose a, b, and c as the identity, thus R is not null... –  Malcolm Lazarow May 28 at 2:52

1 Answer 1

Transitivity: $\forall a,b,c \in A$[$(aRb$ and $bRc) \rightarrow aRc$]

So, let's assume that $aRb$ and that $bRc$. Then $\exists d,f \in A$ such that $a*d=d*b$ (1) and $b*f=f*c$. (2)

We multiply $f$ to both sides of equation (1): $a*d*f=d*b*f$. By equation (2), we can substitute $f*c$ for $b*f$, so $a*d*f=d*f*c$.

If $A$ is closed under $*$, then we can conclude $d*f \in A$ and thus from the above that $R$ is transitive.

I'll keep thinking about symmetry.

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