Suppose T1(N) = O (f (N)) and T2(N) = O (f (N)). Which of the following statements are true in gen

Question

Q Suppose T1(N) = O (f (N)) and T2(N) = O (f (N)).

Which of the following statements are true in general?

 (a) T1(N) + T2(N) = O (f (n))

 (b) T1(N) − T2(N) = o (f (n))

 (c) T1(N) / T2(N) = O(1)

 (d) T1(N) = O (T2 (N))

Comments

Look what have you done is

T₂ <= c₁ (f(n)) .....................................(1)

T₁ <= c₂ (f(n))......................................(2)

=> (T₁/T₂) <= c₂/c_{1 ...................................(3)}

=> T₁ <= (c2/c1) T₂

Let us assume what you have done is right then it should also hold true for integers

So, assume f(n) = 9  T₂ = 3 and T₁ =4 c₁ and c₂ = 1

Then the inequality (1) becomes

3 <= 9  which is true.

the inequality (2) becomes

4 <= 9 which is true. 

=> 4/3 <= 1  (the inequality number 3 according to you) 

IT IS WRONG 

So the operations you have done on inequality is wrong. 

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In asymptotic notation constant are not effect it may be grater then constant.  We have to solve asymptotically not mathematically.  

Here

T1 = n and T2 = n+2 .

Asymptotically T1=o(T2)

But mathematicaly  T1 <T2

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So, I think you are telling that the inequality calculations you have done are correct asymtotically but not mathematically.

So, lets prove it that your calculation is also wrong asymtotically.

So, what you have done is

T₂ <= c₁ (f(n)) .....................................(1)

T₁ <= c₂ (f(n))......................................(2)

=> (T₁/T₂) <= c₂/c_{1 ...................................(3)}

=> T₁ <= (c2/c1) T₂

This time assume T₂ = n , T₁ = n² , f(n) = n³

Now the inequality 1 

T₂ <= c₁ (f(n))

Now, according to my assumption

n <= c₁ (n³)

I think  this is right because n is asymtotically less than n³

Tell me if I am wrong.

Now, come to the inequality 2

T₁ <= c₂ (f(n))

Now, according to my assumption 

n² <= c₂ n³

I think this is right because n² is asymtotically less than n³

Tell me if I am wrong.

Now, come to the inequality number (3)

(T₁/T₂) <= c₂/c₁

Now according to my assumption.

(n²/n) <= c₂/c₁

=> n² <= (c₂/c₁) n

I think this is wrong because n² is not asymtotically less than n

Tell me if I am wrong.