Properties of Asymptotic Notations || GATECSE || DAA

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Asymptotic notations are mathematical concepts used to measure the growth or decay of a function as the argument, denoted by 'n', approaches infinity. Common notations include Big-O, little-O, omega, and Theta, which are used in computer science and mathematical analysis to describe algorithm efficiency and function behavior.
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Properties of Asymptotic Notations:
Reflexivity:
If f(n) is given then
f(n) = O(f(n))
f(n) = Ω(f(n))
f(n) = Θ(f(n))
Symmetry:
f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n))
Transistivity:
f(n) = O(g(n)) and g(n) = O(h(n)) ⇒ f(n) = O(h(n))
f(n) = Θ(g(n)) and g(n) = Θ(h(n)) ⇒ f(n) = Θ(h(n))
f(n) = Ω(g(n)) and g(n) = Ω(h(n)) ⇒ f(n) = Ω(h(n))
f(n) = o(g(n)) and g(n) = o(h(n)) ⇒ f(n) = o(h(n))
f(n) = ω(g(n)) and g(n) = ω(h(n)) ⇒ f(n) = ω(h(n))