Question

Find the big−O, big−Ω estimate for x⁷y³+x⁵y⁵+x³y⁷. [Hint: Big-O, big- Ω, and big-Θ notation can be extended to functions in more than one variable. For example, the statement f(x, y) is O(g(x, y)) means that there exist constants C, k₁, and k₂ such that |f(x, y)| ≤ C|g(x, y)| whenever x > k₁ and y > k₂.]

Answer 1

big−O:

f(x, y) ≤ Cg(x, y)

f(x, y)=x⁷y³+x⁵y⁵+x³y⁷ ≤ 3g(x, y)=3x⁷y⁷

g(x, y)=3x⁷y⁷ , C=2 for $x\ge 1,\ y\ge 1$

big−Ω:

Cg(x, y)≤f(x, y)

3g(x, y)=3x³y³ ≤ f(x, y)=x⁷y³+x⁵y⁵+x³y⁷

g(x, y)=x³y³ , C=3 for $x\ge 1,\ y\ge 1$