Determine if correct or incorrect and justify yourself.
n^3 = O(n^2 + n^3)
The O is the big Oh notation.
To determine if the statement "n^3 = O(n^2 + n^3)" is correct or incorrect, we need to evaluate whether the left-hand side (n^3) is upper-bounded by the right-hand side (n^2 + n^3) as n approaches infinity.
In Big O notation, if a function f(n) is said to be O(g(n)), it means that there exists some constant c and some value n0, such that for all values of n greater than or equal to n0, f(n) is less than or equal to c * g(n).
So, let's analyze the given expression:
n^3 = O(n^2 + n^3)
If we simplify the right-hand side, we get:
n^3 = O(n^3) (since n^2 is smaller than n^3 as n approaches infinity)
For the left-hand side n^3 to be O(n^3), it means that there exist constants c and n0 such that for all values of n greater than or equal to n0, n^3 is less than or equal to c * n^3.
This is trivially true, as n^3 is always less than or equal to c * n^3 for any constant c. Hence, the statement "n^3 = O(n^2 + n^3)" is correct.
In summary, "n^3 = O(n^2 + n^3)" is a correct statement because the left-hand side is indeed upper-bounded by the right-hand side as n approaches infinity.