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Posted 28 February 2008

SETS: METAPHORS AND IMAGES

This section describes certain common images and metaphors we have for sets  and their dangers.

A set is a mathematical object

The set  is a math object.  It is not five different numbers, it is one object and its defining property is that only the numbers 3, 5 ,6, 7 and 8 are elements of the set.  The set  is not the same thing as its elements.  It is true that the integer 5, for example, has a special relationship to the set, namely it is one of the set’s elements, but it is important to think of the set as separate from its elements. 

The set  of all real numbers is a single math object with unimaginably many elements.  You may think it impossibly hard to think of all the real numbers at once.  But  is just one math object.  You don’t have to think about all the real numbers at once to think about , any more than you have to think about all the molecules of water in the Pacific Ocean at once to think about the Pacific Ocean. 

A set “is” its notation

 not!

When you visualize the set , the image you have may be the notation “  ”.  (See list notation.) It is not entirely bad to visualize the notation when you think of this set, but remember that the notation has an irrelevant feature, namely the order in  which the elements are written.  The set  is the same set as the one denoted by .   Similarly, for real x,  even though the notation is different.

The same set can be described by many different notations.

A set is a container

You can think of a set as a container that holds all its elements.   This metaphor is good in some ways and bad in others.

¨  Good: A container is a single object, different from any of the things it contains.  In fact, it is different from the whole bunch of things it contains.

¨  Bad: Consider the sets  and .  Then  and  and in fact .  How can one thing be in three different containers?

¨  Bad:  Using the sets A and B just defined, note that  and .  But .  This behavior is not at all the way we usually think of containers:  If you have a dime in a coin purse and the coin purse is in your pocket, you would say that the dime is in your pocket.

Venn Diagrams

¨  Good:  Venn Diagrams give a useful picture of relationships between a few sets portrayed as all the points inside a circle (or other closed curve).   The circle is then a container. 

¨  Bad:  The elements of the set inside the circle may be shown as points in the plane.  Some are close to each other and one may be above another.  Even so, the location of the points is irrelevant.  

A set is a collection

A set can be thought of as a collection of objects.  But you must think of the collection as an abstract object, not as a bunch of things.  

Example:    The  set of planets in our solar system (however it is defined) is a collection of physical objects, but the collection itself is an abstract idea, not a physical thing.

A set is a pointer

I prefer to think of the set  as a mathematical object that has a special relationship with the numbers 3, 5, 6, 7 and 8 that it does not have with any other objects.  My picture is a node called A that has wires or arrows connecting A to those five numbers and to nothing else.  Those numbers can be moved around in the picture but that makes no difference  all that counts is the connections.  (In that sense this picture is a directed graph.)

If you know something about programming, you may see a similarity between this and a pointer in C or Pascal.  Typically a pointer in those languages will point to the head of a list and each entry in the list will point to another one.  But the proper image for a set is that the pointer points to each one of them without preference  there is no ordering, implicit or otherwise.  Another difference is that for sets, unlike computer language pointers, two different sets cannot point to exactly the same bunch of elements.