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**Properties of Asymptotic Notations**

In this article, I am going to discuss **Properties of Asymptotic Notations**. Please read our previous article where we discussed **Asymptotic Notations**. As part of this article, we are going to discuss the following Asymptotic Notations Properties.

**General Properties****Reflexive Properties****Transitive Properties****Symmetric Properties****Transpose Symmetric Properties**

**Properties of Asymptotic Notations:**

As we have gone through the definition of these three notations (**Big-O, Omega-Q, Theta-Θ**) in our previous article. Now let’s discuss some important properties of those notations.

**General Properties:**

If f(n) is O(g(n)) then a*f(n) is also O(g(n)) ; where a is a constant.

**Example: **

**f(n) = 2n²+5 is O(n²)**

**then 7*f(n) = 7(2n²+5)**

**= 14n²+35 is also O(n²)**

Similarly this property satisfies for both Θ and Ω notation. We can say

If f(n) is Θ(g(n)) then a*f(n) is also Θ(g(n)); where a is a constant.

If f(n) is Ω (g(n)) then a*f(n) is also Ω (g(n)); where a is a constant.

**Reflexive Properties:**

If f(n) is given then f(n) is O(f(n)).

**Example: f(n) = n² ; O(n²) i.e O(f(n))**

Similarly, this property satisfies both Θ and Ω notation. We can say

If f(n) is given then f(n) is Θ(f(n)).

If f(n) is given then f(n) is Ω (f(n)).

**Transitive Properties :**

If f(n) is O(g(n)) and g(n) is O(h(n)) then f(n) = O(h(n)) .

**Example: if f(n) = n , g(n) = n² and h(n)=n³**

**n is O(n²) and n² is O(n³) then n is O(n³)**

Similarly this property satisfies for both Θ and Ω notation. We can say

If f(n) is Θ(g(n)) and g(n) is Θ(h(n)) then f(n) = Θ(h(n)) .

If f(n) is Ω (g(n)) and g(n) is Ω (h(n)) then f(n) = Ω (h(n))

**Symmetric Properties :**

If f(n) is Θ(g(n)) then g(n) is Θ(f(n)) .

**Example: f(n) = n² and g(n) = n² then f(n) = Θ(n²) and g(n) = Θ(n²)**

This property only satisfies for Θ notation.

**Transpose Symmetric Properties :**

If f(n) is O(g(n)) then g(n) is Ω (f(n)).

**Example: f(n) = n , g(n) = n² then n is O(n²) and n² is Ω (n)**

This property only satisfies for O and Ω notations.

**Some More Properties :**

1. If f(n) = O(g(n)) and f(n) = Ω(g(n)) then f(n) = Θ(g(n))

2. If f(n) = O(g(n)) and d(n)=O(e(n))

then f(n) + d(n) = O( max( g(n), e(n) ))

**Example: f(n) = n i.e O(n)**

d(n) = n² i.e O(n²)

then f(n) + d(n) = n + n² i.e O(n²)

3.If f(n)=O(g(n)) and d(n)=O(e(n))

then f(n) * d(n) = O( g(n) * e(n) )

**Example: f(n) = n i.e O(n)**

d(n) = n² i.e O(n²)

then f(n) * d(n) = n * n² = n³ i.e O(n³)

**Commonly used Logarithms and Summations:**

In the next article, I am going to discuss **Master Theorem**. Here, in this article, I try to explain Properties of Asymptotic Notations. I hope you enjoy this **Properties of Asymptotic Notations** article. I would like to have your feedback. Please post your feedback, question, or comments about this article.