# Teaching set theory in NY – more than the minimum

We didn’t used to teach sets in New York State; now we have to – to Algebra students (usually 9th graders).

It is not clear exactly what to teach.

There are only three indicators related to sets.

And the glossaries, indicators, and actual tests (3 of them) define a fairly limited set (ha ha) of items that need be taught:

- what a set is, how to write it (including both roster and set builder notation), what union and intersection mean, and what “complement” means. Also, the symbols for all of these. (for complement they offer prime and superscript C, but neither has occurred on a test)

It would be a shame to teach this topic to this test.

My suggestion is to link the set theory to as much of the rest of the course as possible. Here are some places to do it outside of the unit:

- Solution Sets. As we run into them, {-3, 5}, identify them as
*sets*. Make certain we know they are not ordered. - Natural Numbers, Whole Numbers, Integers, Rational Numbers, Real Numbers. To the extent possible, identify these as sets. For an advanced class, introduce set-builder notation for the rationals as a “gee whiz” wow them sort of thing. It’s actually not so terrible, since the “fraction” and the restriction should look vaguely familiar. (Ironically, simpler examples may be harder to understand)
- Problem Solving (see Subset Puzzle)

When we reach the unit, the concept of subset gets short shrift in the standards. It can be treated more fully. And cardinality, easy and useful, is omitted. It can painlessly be included. And why no empty set? Throw it in.

Here are some ways to draw links to other parts of mathematics and this course in particular, within the unit (I’ve never hit all of them). Special emphasis on differences from and similarities with properties of arithmetic.

- Use the “not” symbols: ∉, ⊄ which kids quickly figure out, and like (since they are analagous to ≠).
- Draw the analogy between union ∪ and addition, but run some compare and contrast.
- Draw the analogy between intersection ∩ and multiplication, but run some compare and contrast.
- Draw the analogy between the complement of a set and the opposite of a number.
- Draw the analogy between “is a subset of” and “is less than”. Enjoy the superficial resemblance: ⊂ and < . (It helps if they know what cardinality means)
- Show that union and intersection are commutative and associative
- Compare the distributive property of multiplication over addition with how distribution seems to work for union over intersection and intersection over union.
- Examine the complement of a union. Compare with the opposite of a sum.
- The union of a set and the empty set is the first set. (compare at addition of 0)
- The intersection of a set and the empty set is the empty set. (compare at multiplication by 0)

wow, i wish i would have read something like this earlier in the year. first year mistakes, i guess… but it’s now may and i’m planning on throwing set notation into a review session at the end of the month.

i always said i’d do better next year!