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Set Inclusion Notation

Date: 9/12/96 at 19:6:56
From: Anonymous
Subject: Set Inclusion Notation

Dear Doctor Math,

When using the defining property version of set notation: {x|x<2}, 
why is the first "x" necessary? Why can't we just say {x<2} to read as 
"the set of all x's less than 2", instead of "the set of all x's such 
that x is less than 2"?

Date: 9/12/96 at 19:12:54
From: Doctor Jodi
Subject: Re: Set Inclusion Notation

Hi there! You're question is a very good one.  I think the problem is
that a set can look like this:

{5, elephant, 7/9, Aegean Sea}

So your set {x<2} could mean that the expression "x<2" is part of the 
set. Though it is tedious to have to write {x|x<2} out, it's important 
to make the distinction clear.

Let us know if you have other questions.

-Doctor Jodi,  The Math Forum
 Check out our web site!   

Date: 9/12/96 at 21:49:10
From: Louis Feicht
Subject: Re: Set Inclusion Notation

Dear Dr. Math,

In the set {5, elephant, 7/9, Aegean Sea}, an elephant is a member 
of the set, not the "e", "l" etc. It is what the word represents, just 
as  x<2 doesn't mean the symbols "x", "<". I'm not sure I understand 


Date: 9/12/96 at 22:49:12
From: Doctor Pete
Subject: Re: Set Inclusion Notation

Here's another way of looking at it.  Say you have the set


Although 5+7 = 12, clearly the purpose in writing the above expression
instead of {12,12} or {12} is to emphasize the addition operation on 
the two numbers 5 and 7.  By the same token, we can also say


and this is quite different than simply saying, {12,12,12}.  Now, most 
of the time we wouldn't be interested in the different ways of writing 
12, but this was just an example of how equivalence doesn't preserve 
information; that is, in the process of replacing 5+7 with 12 you lost 
the original "5" and "7" which is different from "4" and "8".  Here's 
something that more closely illustrates your original question:  What 
is the difference between

     {x and y in Z+ and x+y = 12}


     {x | x and y in Z+ and x+y = 12} ?

The first reads,

     "The set containing the criterion 'x and y are positive integers
      whose sum is 12'"

and the second reads

     "The set of all x such that x and y are positive integers
      whose sum is 12."

There is a difference.  You could interpret the first set in two ways:  
You can choose to evaluate the criterion, or leave it alone.  But if 
you evaluate it, what quantity is it that is required from this 
condition?  x? y?  x+y?  x^2+y^2?  Because this is ambiguous, we can 
only interpret the first set as a single-element set which contains an 
element which is a condition.  On the other hand, the second set is 
unambiguous, and is equivalent to


Does this make sense?  Let me apply the same principle to your 
original statement,

     {x < 2}  and   {x | x < 2} .

The first says "The set containing the criterion 'x is less than 2'" 
and the second says, "The set of all x such that x is less than 2."  
What's the difference?  Well, in the first set, *you cannot assume 
that what you are asking for is x!*  How do you know that what you 
want is x and not x+2?  I could say,

     {x+2 | x < 2},

which under the reals is equivalent to

     {x | x < 4}.

But simply saying {x < 2} tells us nothing of how to apply this 
condition.  It is not right to assume that all we want is x, because 
if that were the case we should have written {x | x < 2}.

Does this make sense?  The point is that trying to apply the condition 
in {x < 2} is not possible unless you know how, which is precisely 
what {x | x < 2} tells you.  Because this ambiguity arises, you must 
treat the first set as a set containing one element which is the 
condition "x < 2". Notice I never treated "x < 2" as something 
composed of pieces, like "x", "<", etc.  It is a condition, a 
restriction as to what values x may take.

-Doctor Pete,  The Math Forum
 Check out our web site!   

Date: 9/13/96 at 13:21:15
From: Louis Feicht
Subject: Re: Set Inclusion Notation

Thanks, that helped a lot! I'm still going to have to think about some 
more examples before I can explain it to eighth graders

Associated Topics:
High School Basic Algebra

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