Following are the properties of asymptotic notations:-

**1. Transitive**

- If f(n) = Θ(g(n)) and g(n) = Θ(h(n)), then f(n) = Θ(h(n))
- If f(n) = O(g(n)) and g(n) = O(h(n)), then f(n) = O(h(n))
- If f(n) = o(g(n)) and g(n) = o(h(n)), then f(n) = o(h(n))
- If f(n) = Ω(g(n)) and g(n) = Ω(h(n)), then f(n) = Ω(h(n))
- If f(n) = ω(g(n)) and g(n) = ω(h(n)), then f(n) = ω(h(n))

**2. Reflexivity**

- f(n) = Θ(f(n))
- f(n) = O(f(n))
- f(n) = Ω(f(n))

**3. Symmetry**

- f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n))

**4. Transpose Symmetry**

- f(n) = O(g(n)) if and only if g(n) = Ω(f(n))
- f(n) = o(g(n)) if and only if g(n) = ω(f(n))

**5. Some other properties of asymptotic notations are as follows:**

- If f (n) is O(h(n)) and g(n) is O(h(n)), then f (n) + g(n) is O(h(n)).
- The function loga n is O(logb n) for any positive numbers a and b ≠ 1.
- loga n is O(lg n) for any positive a ≠ 1, where lg n = log2 n.