Sage handles pre-images just a bit differently than our approach in the text. For the moment, we can obtain a single vector in the set that is the pre-image via the .preimage_representative() method. Understand that this method will return just one element of the pre-image set, and we have no real control over which one. Also, it is certainly possible that a pre-image is the empty set — in this case, the method will raise a ValueError. We will use our running example to illustrate.
Remember, we have defined the pre-image as a set, and Sage just gives us a single element of the set. We will see in Sage ILT that the upcoming Theorem KPI explains why this is no great shortcoming in Sage.