"odd + odd doesn't equal even?"

This subgroup gives us two cosets: $~$0 + 2\\mathbb Z$~$ and $~$1 + 2\\mathbb Z$~$ \(remember that $~$+$~$ is the group operation in this example\), which are the elements of our quotient group\. We will give them their conventional names: $~$\\text{even}$~$ and $~$\\text{odd}$~$, and we can apply the coset multiplication rule to see that $~$\\text{even}+ \\text{even} \= \\text{even}$~$, $~$\\text{even} + \\text{odd} \= \\text{odd}$~$, and $~$\\text{odd} + \\text{odd} \= \\text{odd}$~$\.

odd + odd doesn't equal even?