Assigning Grades for Mathematical Proofs: A, C, or F - Explanation Included

[Jacobpm64]

Homework Statement

Assign a grade of A (excellent) if the claim and proof are correct, even if the proof is not the simplest or the proof you would have given. Assign an F (failure) if the claim is incorrect, if the main idea of the proof is incorrect, or if most of the statements in it are incorrect. Assign a grade of C (partial credit) for a proof that is largely correct, but contains one or two incorrect statements or justifications. Whenever the proof is incorrect, explain your grade. Tell what is incorrect and why.

Claim: If A, B, and C are sets, and $$A \subseteq B$$ and $$B \subseteq C$$, then $$A \subseteq C$$.

Proof: Suppose x is any object. If $$x \in A$$, then $$x \in B$$, since $$A \subseteq B$$. If $$x \in B$$, then $$x \in C$$, since $$B \subseteq C$$. Therefore $$x \in C$$. Therefore $$A \subseteq C$$.

The Attempt at a Solution

Am I correct in wanting to give this proof a grade of an A, even if the language seems a bit shaky?

 

[bob1182006]

Well there are no incorrect statements or justifications right?

You would just have stated it a bit more...rigorous?

It seems like it deserves an A though.

 

[Jacobpm64]

Well I would have said.

Proof: Let $$x \in A$$. Since $$A \subseteq B$$, $$x \in B$$. Similarly, since $$B \subseteq C$$, $$x \in C$$. Hence, $$A \subseteq C$$.

I guess they are the same though.

 

[HallsofIvy]

Jacobpm64, there is a slight problem with your 'way of saying it'. You start by saying "let $x \in A$". What happens if A is empty? You would need to either include a separate (very simple) proof for the case that A is empty or start with "If $x \in A$" as was done in the given proof. That way, if A is empty, the hypothesis is false and the implication is "vacuously true".

 

[matt grime]

I also disagree - as you say yourself, Halls, when the precedent is false the implication is true. How can there be a problem there?

 

[StatusX]

The only thing I might do with Jacobpm64's proof is changing the word "let" to "assume". "Let" suggests the thing you're doing is known to be possible, while "assume" is more hypothetical. But this is just the way these words are usually used (at least in my experience), and I wouldn't take any credit off either proof.