Ok so lets say that the asymptotic execution time of an algorithm B is greater than the asymptotic execution time of an algorithm A .
Τ(Ν)Β = c1*f(N)B > Τ(Ν)A = C2*f(N)A for c1 and c2 that make the equals statement true
Can we say that as n(input) increases , A will run faster than B?
I'm new here , so forgive me if I got the tags wrong, thanks for your time!
If the execution time of $A$ is asymptotically smaller than $B$, it means that for sufficiently large input size $n$, $A$ is faster than $B$. More rigorously, there exists a positive integer $N$ such that, for all input size $n\geq N$, $A$ runs faster than $B$.