Let , and .

Could someone please show me how to find the cosets of in , and the cosets of in ?

This is the answer given in the book, but I don't know how to get this answer:

Of course, I know by Lagrange's theorem that there must be cosets in both groups. In for each coset we use the notation , where is called the coset representative of . So how do we go about choosing these coset representatives?

Can't we just choose any element from from ? Why did they choose those specific elements for those cosets of each group?