Set builder notation and proofs

[Stephen Tashi]

"set builder" notation and proofs

I'm curious about the references to "set builder" notation that I see in forum posts. Is this now a popular method of teaching elementary set theory and writing elementary proofs?

I haven't looked at materials for that subject in the past 20 years. The use of the "A = {x:...}" type of notation is very old, but in some posts I see that people are encouraged to write proofs that consist exclusively of "steps" that use that notation. Unlike the old fashioned "steps and reasons" form of proof used in secondary school, these "set builder" "proofs" often don't give a reason for each step.

The specialized methods used by the educational community to do proofs are compromises between two goals. 1. Having the student demonstrate an understanding of the material. 2. Using a format that is easy to grade ! These two goals are somewhat at odds with each other. Things that are easy to grade tend to be abbreviated and sometimes students who don't competely understand what they are doing can still write down the right symbols. However, compromises must be made in teaching classes. It seems to me that proofs in "set builder" notation are such a compromise.

 

[HallsofIvy]

"Set builder notation" is neither a method of teaching set theory nor a means of proof. It is, just as the name says, a notation of writing and specifying sets. "The set of all even numbers" would be written, in set builder notation, as {x| x=2n, n an integer}. That would be read as "the set of all x such that x is 2 times an integer".

 

[Stephen Tashi]

"Set builder notation" is neither a method of teaching set theory nor a means of proof.

The would be opinion of my generation. In fact, I don't recalled seeing the terminology "set builder notation" in any textbook. I'm merely curious whether modern textbooks have begun to use that terminology and emphasize it.

 

[HallsofIvy]

I remember the precise phrase "set builder notation" in algebra and precalculus book back to 20 years ago.

 

[Hootenanny]

I have also seen a few books that I wouldn't consider "modern" using set builder notation and the associated terminology. A proof must stand on its own, as it where, and be understandable in isolation. Perhaps you are referring to cases where the "steps" themselves are sufficient to understand the proofs without requiring further elucidation.

 

[Stephen Tashi]

I remember the precise phrase "set builder notation" in algebra and precalculus book back to 20 years ago.

That would be be long after I was reading such books!

 

[Stephen Tashi]

A proof must stand on its own, as it where, and be understandable in isolation. Perhaps you are referring to cases where the "steps" themselves are sufficient to understand the proofs without requiring further elucidation.

In education, the standard for what can be done without further elucidation is quite variable and heavily influenced by the goal of making proofs easy to grade. (For example, the old fashioned way of "proving" trigonometric identities begins with two non-identical expressions set equal, or set equal with a question mark over the equal sign. It proceeds to use "steps" to arrive at two identical expressions. A real proof would have to reverse the process. Not all mathematical "steps" are reversible and a real proof would have to demonstrate the reasoning worked in the correct order. However, it is far easier to grade work that is done the old fashioned way and students like it better since it involves less labor.)

Specifically, I am curious whether some instructors teach a format for proofs in elementary set theory that abbreviates most of the work to manipulations of the "set builder" notation for sets. I see various posts about set theory homework questions, where the poster has attempted to write such an abbreviated proof. I don't know if this shows a common type of confusion that afflicts students or whether it shows attempts to imitate a style that has been taught in class.