Many theorems have the form : Premise(es) implies Conclusion(s)


**Example A of wrongness**:  

There are many examples in which a theorem is stated without mentioning that part of the premise is not necessary to reach the conclusion.  

Usually it is simple (and much better) to add a remark stating that the result is not sharp (ideally providing an example of weaker premise holding with the solution).


But there is another type of bias : 


**Added Note:** Below composition means the AND of two relations ( for classical composition the transitivity does not compose! ( thanks to HenrikRüping remark). 


**Example B of wrongness**:  

*Theorem 1* : The composition of 2 equivalence relations is an equivalence relation.  

Or in fewer words : Equivalence relations are stable under composition.


Actually there is a much finer version of B : 


*Theorem A*: For relations each of the following properties are stable under composition : Reflexive , Transitive , Symmetric.  

By conjunction of the above we obtain:  

*Corollary B*: Equivalence relations are stable under composition


*Note:* The second form is not only more precise but it also makes the mention "left as an easy exercise" more acceptable.


The "WRONG" notion:  

I called theorem 1 (or its statement) **wrong** as it induced the reader to think that the conjunction of the 3 properties plays a role in proving the conclusion. 


Of course only true theorems may be qualified as wrong. 


Taking an absolute stance you may call wrong any theorem that is not a tautology.  

A less absolute stance would call wrong any theorem that is not a tautology and in which you forget to mention non-sharpness. 


**Question 1:** is there a better / more adequate term than wrong ( the subtext is: do you think it is a good notion?) .


**Question 2:** Do you know examples that follow a pattern like B or some variation in lack of tautology? 


ADDED TO BE MORE SPECIFIC: 


**Question 3:** 
More specifically : Are there other types of patterns showing a distance between premise and conclusion. The types need to be common in the mathematical literature, not purely logical types ( of course those are more countable). 



Theorems true but wrong.
A classic example for B is the theorem (proved using Gauss' lemma) usually stated as: if $R$ is a unique factorization domain, then so is the ring of polynomials $R[x]$. Now, $R$ is a UFD iff it is a GCD-domain with no infinite descending chain of proper divisors (ACCP), and


* if $R$ is a GCD-domain, then so is $R[x]$,
* if $R$ satisfies ACCP, then so does $R[x]$.


