Q:

Is this understanding of sufficiency correct?

If X is a sufficient statistic for $\theta$, I know that any g(X) where g(X) is one to one is also a sufficient stat. However, does that mean that (X,Y) is also a sufficient stat for $\theta$? 
On the one hand, I'm thinking it is because semantically, since X is "sufficient" doesn't that mean that adding more to it will not diminish its being sufficient?
On the other, I'm thinking it is not because from the definition of sufficiency, it is possible that evaluating the conditional probability will not yield an expression that is independent of $\theta$ when Y is added to the mix.

A:

Yes if you add information from the data in addition to a sufficient statistic you will still have a sufficient statistic.  However, it is a minimal sufficient statistic that should be sought because the idea is to use the statistic that is smallest. For example the sample mean from a normal distribution with known variance is a minimal sufficient statistic for the population mean.

