The students are aware of mathematical logic and proof but have not come across any of the notions of a set. What is the most natural and motivating way to introduce set theory?
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1$\begingroup$ I think it would help potential answerers to answer your question if you expanded your question to describe the class in which "the students" you refer to are currently enrolled. (A) If you are teaching a discrete math or discrete structures course and you need to include a brief introduction to introductory set theory, there are a few great texts; (B) however, if your "the students" are enrolled in a semester's course in introductory set theory, there are other texts that best serve this purpose. But the intersection of recommended texts for (A) and (B) do not intersect. $\endgroup$– amWhyApr 15, 2019 at 16:57
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$\begingroup$ Also, it seems to me that your students might be aware of propositional and predicate logic, and have some experience doing proofs. If they studied mathematical logic, they would have invariably covered introductory set theory in the said course. E.g., the text "A mathematical introduction to Logic" by H. Enderton devotes chapter 0 to basics about sets. $\endgroup$– amWhyApr 15, 2019 at 17:08
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$\begingroup$ What aspect are you worrying about? If they somehow already know logic and proofs, why not just get on with it? Is there some particular point you expect them to have problems with? $\endgroup$– Jessica BApr 15, 2019 at 18:09
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$\begingroup$ This is part of the introductory to number theory. I would ideally like to connect this to logic and proof rather than a set of isolated rules. I am looking for a narrative. $\endgroup$– matqkksApr 15, 2019 at 18:26
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1$\begingroup$ This is extremely vague and overly broad. $\endgroup$– user507Apr 16, 2019 at 2:38
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The student background in proofs and logic is not necessary for initial explanation/motivation of the concepts. And actually it worries me that you mention that when wanting to know how to do an initial introduction! I remember learning the basics of sets (and union, intersection, etc.) as an elementary schoolkid in the New Math. You do not need a huge background to start looking at the topic. This is very different from calculus where you really do need some background in algebra, functions, analytic geometry, etc.
The easiest way to first explain the concept is as a collection of things (e.g. stamps, books, etc.) see pages 5-6 of Tenenbaum Ordinary Differential Equations.
Of course you should move to generalize to infinite sets and more difficult concepts. And the students have background to handle that. But to START, just do exercises with simple sets of integers. E.g. (1, 3, 5, 7, 9) versus (2, 4, 6, 8, 10) versus (3, 7, 11, 214) versus (3, 5, 7, 1, 9). I recommend using integer collections as the starting point as it is easy to look at them and discuss union/intersection. Of course, you can move to more fancy complicated things. But start simple.
You can illustrate with Venn diagrams also.
Of course, given the level of the students you can move to some formal proofs as well. AFTER they have been introduced to the topic. but I would not START any topic with a definition, theorem, proof, example style of teaching.
P.s. I also suggest looking at some books (hard and easy) for their approaches. This should not just be something to be solved de novo by a question here. [Or with an approach of constructing a course from scratch. Anyone who DOES construct a course from scratch should have some deep perspective before doing so...and I don't mean deep math perspective. I mean in addition to that having seen different pedagogical approaches and having some formed ideas on what works/doesn't.]
