Calculating composition of permutation

Question

I'm trying to understand how composition of permutation works on other functions.

I do understand something simple like this

F = (1 2 3 4) G = (1 2)(3 4)

Therefore G σ F = FG = (1 2 3 4)(1 2)(3 4) = (2 4)

Reason for the answer above is

1. 1 maps 2, 2 maps 1, thus 1 maps 1
2. 2 maps 3, 3 maps 4, thus 2 maps 4
3. 3 maps 4, 4 maps 3, thus 3 maps 3
4. 4 maps 1, 1 maps 2, thus 4 maps 2

Therefore, FG = (2 4)

So for another two examples

1. (1 2)(1 0 2) Answer : (0 2) //given by my lecture notes

2. (0 1)(2 3)(0 1 2 3) Answer : (0 2) //given by my lecture notes

This is how i try to calculate for example 1.

1. 1 maps 2, 2 maps 1, thus 1 maps 1

2. 2 maps 1, 1 maps 0, thus 2 maps 0

Therefore my answer for example 1 is (2 0). However, can i say that (2 0) is the same as (0 2)?

However, the method i used for the first 2 examples does not seem to work the same way as example 2. Is there any other way i can calculate that out ?

P.S : I'm also a little curious. Does the "formula" G σ F means FG? So does that means that A σ B = BA?