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2 added 1 characters in body

To complement some of the previous answers, notably the one by Zach Cont.

For question 1, mostly for finite sets, but not only: At the most elementary level as at the most involved or most internal to mathematics, set theory, classical logic and combinatorics are deeply related. Most applications of any two of them could be seen actually as an application of the third.

Many use of diagrams outside mathematics use a combination of naive set theory and classical boolean logic. The technical language of many disciplines use the terms union, intersection, complement of sets, and use a correspondence between (logical) combination of conditions on elements and combination of subset creation and operations. This can be traced at least to Leibniz and probably to medieval times (scholastic tradition for instance). Traditional names in this area are mostly from the 19th such as De Morgan, Boole, Pierce, GrassmanGrassmann, Venn, Cayley.

In this context, it makes sense to study more precise treatment of set theory so that it reinforces intuition of what is reasonable and expressible in this context. It gives clean conceptual tools and refined language to analyze problems and reports opinions and facts. Usually with high school students, the discussion of the classical paradoxes such as Russel and the axiom of foundation leads to better appreciation for the art of defining and for the way to use (even in non mathematical contexts) informal quantifiers and adverbs such as all, always, never, none, nobody, everywhere, everytime, at least, etc.

This might not look very spectacular, but when one considers the usual sloppiness (sometimes voluntary) in newspapers, books and general conversation, this strikes me as very practical for non-mathematicians.

Sound notions of set theory, and the ability to think in terms of cartesian products, relations as quotients, etc. are the basis of a good grasp of probability (see measure theory in other answers) and statistics (and basically experimental data measurement, quantum physics, actuarial techniques, and from the 1960s data mining, database query languages, ...). I certainly do not rule out that we could have developped similar science and technologies by other roads, but it would have given them a very different aspect and to learn all these subjects (instead of recreating them with other foundations) without knowing set theory is especially difficult and limiting for the learner.

1

To complement some of the previous answers, notably the one by Zach Cont.

For question 1, mostly for finite sets, but not only: At the most elementary level as at the most involved or most internal to mathematics, set theory, classical logic and combinatorics are deeply related. Most applications of any two of them could be seen actually as an application of the third.

Many use of diagrams outside mathematics use a combination of naive set theory and classical boolean logic. The technical language of many disciplines use the terms union, intersection, complement of sets, and use a correspondence between (logical) combination of conditions on elements and combination of subset creation and operations. This can be traced at least to Leibniz and probably to medieval times (scholastic tradition for instance). Traditional names in this area are mostly from the 19th such as De Morgan, Boole, Pierce, Grassman, Venn, Cayley.

In this context, it makes sense to study more precise treatment of set theory so that it reinforces intuition of what is reasonable and expressible in this context. It gives clean conceptual tools and refined language to analyze problems and reports opinions and facts. Usually with high school students, the discussion of the classical paradoxes such as Russel and the axiom of foundation leads to better appreciation for the art of defining and for the way to use (even in non mathematical contexts) informal quantifiers and adverbs such as all, always, never, none, nobody, everywhere, everytime, at least, etc.

This might not look very spectacular, but when one considers the usual sloppiness (sometimes voluntary) in newspapers, books and general conversation, this strikes me as very practical for non-mathematicians.

Sound notions of set theory, and the ability to think in terms of cartesian products, relations as quotients, etc. are the basis of a good grasp of probability (see measure theory in other answers) and statistics (and basically experimental data measurement, quantum physics, actuarial techniques, and from the 1960s data mining, database query languages, ...). I certainly do not rule out that we could have developped similar science and technologies by other roads, but it would have given them a very different aspect and to learn all these subjects (instead of recreating them with other foundations) without knowing set theory is especially difficult and limiting for the learner.