Question

Let A and B be sets of real numbers such that A ⊂ B. Find a relation among inf A, inf B, sup A, and sup B.

Let A and B be sets of real numbers and write C = A ∪ B. Find a relation among sup A, sup B, and sup C.

Let A and B be sets of real numbers and write C = A ∩ B. Find a relation among sup A, sup B, and sup C.

Answer 1

1) Since sup B is an upper bound of B and A ⊂ B, it follows that sup B is an upper bound of A, so sup A ≤ sup B, similarly, inf B is a lower bound of B and A ⊂ B, it follows that inf B is a lower bound of A, so inf A ≥ inf B.

2) C = A ∪ B is unbounded above if and only if A is unbounded above or B is unbounded above, in which case we get ∞ = ∞, which is fine. Next, assume A ∪ B is bounded above (and thus A and B are also bounded above.) Since A ⊆ A ∪ B and B ⊆ A ∪ B, we have sup A ≤ sup(A ∪ B) and sup B ≤ sup(A ∪ B), i.e. max{sup A, sup B} ≤ sup(A ∪ B). Finally, note that max{sup A, sup B} is an upper bound of A ∪ B, since x ∈ A implies x ≤ sup A ≤ max{sup A, sup B} and x ∈ B implies x ≤ sup B ≤ max{sup A, sup B}. But sup(A ∪ B) is the least upper bound of A ∪ B, so sup(A ∪ B) ≤ max{sup A, sup B}. Combining without previous inequality, we have sup(A ∪ B) = max{sup A, sup B}, i.e. sup C = max{sup A,sup B}.