The original question mentions beginning students who are defeated by such statements as $\ker(AB) \supseteq \ker(B)$, and I suppose that it is this student population that the original poster ACL is concerned with, rather than more advanced students.
Such students need a very little bit of set language (not axiomatic set theory). And they need a very little bit of formal logic: logical connectives and quantified sentences. Maybe 10 pages altogether of what used to be chapter 0 in every more or less advanced math text. And they ought to be shown how to translate between set operations and logical connectives, e.g. AND $\leftrightarrow \cap$.