Definition of Equal and Equivalent Sets

Date: 05/31/2004 at 15:48:39
From: Sherri
Subject: Math 215

My question is about sets.  I looked through the archives, but I just
can't seem to find the answer.  I'm being asked to write a set that is
equivalent to, but not equal to, the set (a, b, c, d, e, f).  Would
the answer be all the same letters, just arranged in a different order?

Date: 05/31/2004 at 23:12:47
From: Doctor Peterson
Subject: Re: Math 215

Hi, Sherri.

No, the order of the elements in a listing of a set is irrelevant; 
the sets {a,b,c} and {c,b,a} are the same set (equal) because they 
both contain the elements a, b, and c.  That is, the elements of a 
set don't really have any order; you can think of a set as a bag, so 
that it doesn't matter in what order you put objects into it, as 
long as they get in somehow.  Once they are in, you can't tell what 
order they arrived in.

Sets are called equivalent when they have the same cardinality (number
of elements).  What's happening here is that when we ask whether two 
sets are equivalent, we are ignoring the names of the elements, and 
considering only what makes one set essentially different from
another, which is the number of elements it has.  So any set of six 
elements is considered equivalent to the one you were given.

In other words, your question is really about the definition of
'equal' and 'equivalent' in this context.  Equivalent sets contain the
same number of elements, while equal sets contain not only the same
number of elements, but also the exact same elements, regardless of order.

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

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