Big-O, big-Theta, And big-Omega notation can be extended to functions in more than one variable. For example, the statement is means that the exist constants C, k1 , and k2 such that whenever and .

Show that is O(xy).

Step-1:

In this problem we need to show that

Note:

Let us consider f and g are functions from the set of integers to the set of real numbers.

The estimate value can be said that f(x) is O(g(x)) if there are constants C and k such that

, where C > 0 and x> k.The constants C and k are called the witnesses to the relationship.

The definition of f(x) is O(g(x)) says that f(x) grows slower than some fixed multiple of g(x) as x grows without bound.

Step-2:

Consider ,

We...