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What are some applications of sets & relations in science/business/tech that a highschooler can understand?

To kindle a young mind, what examples can be given?

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There probably aren't any. The application of sets and relations is that by studying these things you learn how to reason, and reasoning comes in handy in science etc. – Gerry Myerson Jan 14 '12 at 10:57
I'll throw out a similarly strong contrary position and say there are many applications, particularly in the area of programming as I've detailed a bit below. – dls Jan 15 '12 at 4:13

Well, I would not say there are none, but it is understandably naive to expect that there are going to be direct applications which the «young mind» will be able to understand. From such a basic notion as «sets and relations» to an actual real life application there are layers and layers of abstraction, of concrete application of abstract ideas, and what not. To pick a random example from an answer below, it is rather silly to say that digital electronics is an application of sets and relations—like saying that the Golden Gate bridge is an example of the applications of water, which was though surely involved in many ways in the construction of the bridge.

Sets and relations are part of the background on which modern mathematics is built, and as such takes part in essentially everything we do. So if you want examples point to your cellular phone, to the traffic lights, to the search box in eBay, to the stock market, to a satellite, &c.

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I didn't exactly say that Mariano. – Doug Spoonwood Jan 15 '12 at 3:31
Mostly I agree with this, but relational databases may be a genuine example. – Brian M. Scott Jan 15 '12 at 17:57

Knowing a little bit of set theory is extremely useful for learning SQL. I'm sure the converse is also true. There are many high school aged children who understand the basics of database design, having interest in things like mobile phone application development.

I just read this interesting blog post from Prof Matt Might. He makes a very clear connection between shell programming and set theory. If you're doing non-scientific software development, the type of thinking gained by studying relational algebra (or even just set theory) is invaluable. It also appears in large data-processing programs as the map-reduce pattern.

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One might also find it noteworthy to mention that (classical) set theory has a deep connection with Boolean Algebra in that the algebraic properties of (classical) sets, and the identities of Boolean Algebra match exactly. Boolean Algebra has applications all over the place in digital electronics (almost surely your personal computer got built on this basis). So, one might point out that something very related to set theory has heavily practical applications.

A little more concretely, one might point out the simplification of circuits which happens in Boolean Algebra, but explain that since you aren't going to talk about Boolean Algebra, you'll do such a simplification using the algebra of sets. You just relabel OR gates as union (or UNION if you like) gates, AND gates as intersection gates, and NOT gates as complement gates. Then you write a corresponding set-theoretical expression for such a circuit which seems straightforward, and then simplify it using the algebra of sets. In this way you can make simpler circuits which do the same things as more complicated circuits do by applying set theory.

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Fuzzy subset theory has applications in process control, washing machine technology (well, that does come as process control), expert systems, artificial intelligence, psychology, and a whole host of other fields.

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Groups are used to calculate Hamming distance (used for coding of data). Modulo 2 arithmetic is used for calculating hamming distance between the two codes. This modulo 2 arithmetic is a group as

it is Associative

Has unique identity element 0

And all elements have inverse

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