Tuesday, April 27, 2010

Sets as Plastic Baggies

Is this the picture of an empty baggie?Many times I've seen the use of set-braces (roster notation) compared to an "envelope". Here's one example (speaking of empty sets and the null-set symbol):
To help clarify this concept, think of a set as an envelope. If the set is empty, then the envelope is empty. On the other hand, if the set is not empty -- that is, it contains at least one element -- then there are items in the envelope. One such item can be another envelope. Using this analogy, the symbol {Ø} specifies an empty envelope contained within another envelope. [Setek and Gallo, "Fundamentals of Mathematics" 10th Ed., p. 74]
Now, what I think is the jarring discordance in this analogy: You can immediately see the contents inside a { } symbol, but not so with envelopes (being opaque and all). That's probably why the whole metaphor always feels foul in my mouth, and might be part of the reason I get a poor reaction from students when I try to use it in class.

Let's try a better metaphor: A clear plastic baggie (with a zip, perhaps?). These you can, like the set { } symbol, instantly see into. If you put one plastic baggie inside another, then you can immediately see it sitting inside there... just like the frequently-misused {Ø} notation.

So let's use a metaphor that shares in the transparency and clarity of our precise math notation.

No comments:

Post a Comment