. (a) Prove or disprove carefully and in detail: (i) Θ is transitive and (ii) ω is transitive. (b) Assume n is a positive integer. Algorithm A(n: int) int z = 0; int temp = 0; for (int i = 0; i < n ; i++) { for (int j = 0 ; j < i ; j++) { int temp = i + j ; int z = z + temp; } } System. out. println(z) ; What is the input size for Algorithm A? Analyze carefully and explicitly the time complexity of A in terms of the input size. Show all steps. Is A a polynomial time algorithm? Justify your answer.

The Following are the solution to this question:

Explanation:

In Option a:

In the point (i) is transitive, which means it converts one action to others object because if indicates . It's true by definition, that becomes valid. But if , which implies . it's a very essential component. If . They     will also be .  

In point (ii), The  value of is convergent since the . It means they should be dual a and b constant variable, therefore could only be valid for the constant variable, that is  .

In Option b:

In this algorithm, the input size value is equal to 1 object, and the value of  A is a polynomial-time complexity, which is similar to its outcome that is . It is the outside there will be a loop(i) for n iterations, that is also encoded inside it, the for loop(j), which would be a loop. All internal loops operate on a total number of generations and therefore the final time complexity is .