Rule #1
This rule states that multiplicative constants can be omitted:
c⋅f⪯f.
Examples: 5n2⪯n2, 3n2⪯n2, 7n⪯n.
Rule #2
This rule states that out of two polynomials, the one with a larger degree grows faster:
na≺nb for 0≤a<b
Examples: n≺n2, n≺n32, n2≺n3, n0≺n.
Rule #3
This rule states that any polynomial grows slower than any exponential:
na≺bn for a≥0, b>1
Examples: n3≺2n, n10≺1.1n.
Rule #4
This rule states that any polylogarithm—that is, a function of the form (logn)a—grows slower than any polynomial:
(logn)a≺nb for a,b>0
Examples: (logn)3≺n, nlogn≺n2
Rule #5
This rule states that smaller terms can be omitted:
f⪯g, then f+g⪯g.
Examples: n+n2⪯n2, n9+2n⪯2n.