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).