Which statement is true about Big O, Omega, and Theta notations?

• A. Theta is always larger than O
• B. Omega is always an upper bound
• C. Theta is a tight bound that implies both O and Omega
• D. O is a tight bound

Answer: (C)

A tight bound means the growth of a function is sandwiched between two constant multiples of another function, for large enough inputs. If a function f is in Theta(g), there exist positive constants c1, c2 and n0 such that c1·g(n) ≤ f(n) ≤ c2·g(n) for all n ≥ n0. That exact sandwich shows f grows at the same rate as g up to constant factors, which automatically places f in O(g) (an upper bound) and in Ω(g) (a lower bound) as well. So Theta captures both sides of the bound, making it the tight description. The other ideas don’t hold in general: O is just an upper bound and doesn’t guarantee a matching lower bound, Omega is a lower bound and not an upper bound, and Theta isn’t about being “larger” than O; it’s the precise tight bound that implies both O and Ω.