Don't really understand the notation

Question

I don't really understand how or what I'm supposed to prove. I've researched into each, but still unclear to me what is expected.

Which of the following statements are true? Prove your answers.

1. n² ∈ O(n³)
2. n² ∈ Ω(n³)
3. 2ⁿ ∈ Θ(2ⁿ⁺¹)
4. n! ∈ Θ((n+1)!)

Any help would be much appreciated!

Answer 1

Since this (probably homework) questions is some days old, I think I can answer this question in short.

The wikipedia page (and hopefully your textbook and/or notes too) says

f(n) ∈ O(g(n))     ⇔     lim sup |f(n)/g(n)| < ∞
f(n) ∈ Ω(g(n))     ⇔     lim sup |f(n)/g(n)| > 0
f(n) ∈ Θ(g(n))     ⇔     f(n) ∈ O(g(n)) and f(n) ∈ Ω(g(n))

To prove the left side you can prove the right side.

1. n² ∈ O(n³) is true, due to

   lim sup |n²/n³| = lim (n²/n³) = lim (1/n) = 0 < ∞
   

2. n² ∈ Ω(n³) is false, due to

   lim sup |n²/n³| = lim (n²/n³) = lim (1/n) = 0
   

3. 2ⁿ ∈ Θ(2n+1) is true, due to

   0 < lim sup |2ⁿ/2<sup>n+1</sup>| = lim (2ⁿ/(2⋅2ⁿ) = lim (1/2) = 1/2 < ∞
   

4. n! ∈ Θ((n+1)!) is false, due to

   lim sup |n!/(n+1)!| = lim (n!/((n+1)⋅n!) = lim (1/(n+1)) = 0
   

Notice: All limits holds for n → ∞.