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I recently learned that one of my cousins who is a 7th grader in the United States is learning naive set theory (sets, maps, 1-1, onto maps, relations). Is this part of middle school math curricula in the US? So would your average American high school graduate know what a bijection is?

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Not if they use A Beka Books. Usually US grads know a little set theory, enough to deal with domains, ranges, and solution sets and with inverse functions such as in trig. and log/exp. functions. They are usually familiar with some terms, such as union, intersection, 1-1, onto, but they are not usually familiar with the concept of a bijective function. –  Michael E2 Jan 7 '13 at 21:39
I wouldn't be surprised if there are schools experimenting with such concepts. For example Teaching Set Theory to kids in Middle School: ideas?. Also How to Teach Set Theory to Grade 6. I am not sure what level they are doing, but given the right teacher and students, the kids can amaze. Regards –  Amzoti Jan 7 '13 at 21:40
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2 Answers

At present, there is no standard US curriculum. The mathematics taught before college in public schools varies by the state, the district, the city, the school, and even the classroom.

There is a movement to create national standards ("Common Core State Standards for Mathematics") which have been adopted in most of the states, but they do not presently have a corresponding curriculum.

Historically speaking, you might be interested in reading about New Math, which included a fair amount of naive set theory in its approach to mathematics curricula. Another term worth looking up is SMSG.

Generally speaking, however, the answer to both your questions is no. Naive set theory is not a standard topic in public schools, and the term "bijection" is almost certainly unknown to the vast, vast majority of high school students in the United States.

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I was already beyond all of them, but I actually had a complete set of SMSG draft textbooks around 1961, when I was in junior high; there were some good things in those books. –  Brian M. Scott Jan 7 '13 at 21:50
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In my own (junior) high school (between $5$ and $8$ years ago), there was a mild introduction to these things, but nothing at all on the order of even undergraduate mathematics. As an example, it amounted to having sets $A=\{1,2,3,4\}$ and $B=\{1,2,3\}$ and asking if there could exist a $1$-$1$ map, etc. (This was in a public school, by the way).

From students I've tutored in the past, there also seems to be a similar curriculum, but again, I emphasize that it was not anything extraordinary. This seemed a little long for a comment, and this is my own personal experience. I hope it goes somewhat toward answering your question.

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