words such as "element" and "combining operation" are very general, and can be applied to many real-world and abstract situations. any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). integer numbers, and the operations of addition and multiplication, are just one example. for example, the elements might be computer data words, where the first combining operation is exclusive or and the second is logical conjunction. theorems of abstract algebra are powerful because they are general; they govern many systems. it might be imagined that little could be concluded about objects defined with so few properties, but precisely therein lay noether's gift to discover the maximum that could be concluded from a given set of properties, or conversely, to identify the minimum set, the essential properties responsible for a particular observation. unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions. in his obituary of noether, her student van der waerden recalled that