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**Jacobpm64** 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 $\displaystyle A \subseteq B $ and $\displaystyle B \subseteq C $, then $\displaystyle A \subseteq C $.

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

I do not know which grade to give this proof. It looks like it has the right idea, but its wording seems to be a bit shaky. I was told by my instructor not to use "if, then" statements within a proof. Also, the usage of therefore twice seems a bit shaky. Is this enough to give the proof a C? Besides the shakiness in words, it seems as though the proof as the correct idea. The person assumed the hypothesis and derived the conclusion (a typical direct proof).

Thanks in advance.