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PR2 - Theorem 1 Properties of O Ω θ o ω Proof P1 1 Prove...

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Unformatted text preview: Theorem 1. Properties of O , Ω , θ , o , ω Proof. P1 1. Prove f ∈ O ( g ) and g ∈ O ( h ) ⇒ f ∈ O ( h ). f ≤ c 1 g, g ≤ c 2 h, c 1 , c 2 ∈ R + f ≤ c 1 g ≤ c 1 c 2 h = c 3 h, c 3 ∈ R + ∴ f ∈ O ( h ) 2. Prove f ∈ Ω( g ) and g ∈ Ω( h ) ⇒ f ∈ Ω( h ). f ≥ c 1 g, g ≥ c 2 h, c 1 , c 2 ∈ R + f ≥ c 1 g ≥ c 1 c 2 h = c 3 h, c 3 ∈ R + ∴ f ∈ Ω( h ) 3. Prove f ∈ Θ( g ) and g ∈ Θ( h ) ⇒ f ∈ Θ( h ). By 1 and 2, f ∈ O ( h ) , f ∈ Ω( h ) . ∴ f ∈ Θ( h ) 4. Prove f ∈ o ( g ) and g ∈ o ( h ) ⇒ f ∈ o ( h ). By 1, 2 and 3, f ∈ O ( h ) , f ∈ Θ( h ) . ∴ f ∈ o ( h ) 5. Prove f ∈ ω ( g ) and g ∈ ω ( h ) ⇒ f ∈ ω ( h ). By 1, 2 and 3, f ∈ Ω( h ) , f ∈ Θ( h ) . ∴ f ∈ ω ( h ) P2 1. Prove f ∈ O ( g ) ⇔ g ∈ Ω( f ). If f ∈ O ( g ) , f ≤ c 1 g, c 1 ∈ R + . Then g ≥ 1 c 1 f = c 2 f, c 2 ∈ R + ....
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PR2 - Theorem 1 Properties of O Ω θ o ω Proof P1 1 Prove...

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