Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify as...

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.

(a) Claim. Let A be a set with a partial order R. If C ⊆ B ⊆ A and sup(C) and sup(B) exist, then sup(C) ≤ sup(B).

“Proof.” sup(B) is an upper bound for B. Therefore, sup(B) is an upper bound for C. Thus sup(C) ≤ sup(B).

(b) Claim. Let A be a set with a partial order R. If B ⊆ A, u is an upper bound for B, u ∈ B and then sup(B) exists and u = sup(B).

“Proof.” Since u ∈ B, u ≤ sup(B). Since u is an upper bound, sup(B) ≤ u. Thus u = sup(B).

(c) Claim. For A, with the usual ≤ ordering, sup(A ∪ B) = sup(A) + sup(B).