Sets and Elements

Date: 03/29/2003 at 14:02:57
From: Nanda
Subject: Set

Find the elements of the Set A={{1,2,3},{4,5},{6,7,8}} and determine 
whether each of following is true or false:

   a) 1EA
   b} {1,2,3}CA
   c) {6,7,8}EA
   d) {{4,5}}CA
   e) empty EA
   f) empty CA

I believe that 1,2,3,4,5,6,7,8 are elements of set A. {1,2,3},{4,5},
{6,7,8} are subsets of set A and also elements of set A.

Date: 04/01/2003 at 16:53:37
From: Doctor Peterson
Subject: Re: Set

Hi, Nanda.

I assume you are using "E" to mean "is an element of" and "C" to 
mean "is a subset of."

You have to take the definition of A literally. It does not say that 
1, 2, 3, and so on are elements of the set; if they were, you would 
be told

    A = {1,2,3,4,5,6,7,8}

Rather, you are told that the elements of A are the SETS {1,2,3}, 
{4,5}, and {6,7,8}. A set is not its elements. And a subset is a SET 
of elements, not an element itself.

So, for example, 1 is not an element of A, but {1,2,3} is, and {1,2,3} 
is not a subset of A, but {{1,2,3}} is, since the latter is the subset 
whose only element is the set {1,2,3}, which is in A.

I know it sounds strange, and normal people don't talk this way; but 
math is all about precision - that's why it's written in symbols and 
carefully defined words, rather than in ordinary English. Set 
notation was invented to clarify this sort of thing, so that you can 
say EXACTLY what you mean. That allows us to talk about sets of sets 
without being entirely incomprehensible.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/