Produced by Charles Wells Revised 2016-07-23 Introduction to this website website TOC website index blog Back to head of sets chapter

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

The set $A:=\{3,5,6,7,8\}$ is a mathematical object.

- $A$ is
*one object*. It is not five different numbers. - Its defining property is that the numbers $3$, $5$, $6$, $7$ and $8$
are elements $A$ and
*nothing else*is an element of $A$. - $A$
*is not the same thing as its elements.*It is true that the integer $5$, for example, has a special relationship to $A$, namely it is one of the elements of $A$, but the set $A$ is an object that is*separate*from its elements.

The set $\mathrm{R}$ 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 $\mathrm{R}$ is *just one math object.* You don’t have to think about
all the real numbers at once to think about $\mathrm{R}$, 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
one mathematical object:

not many, not a bunch, but one object.

When you visualize the set $\{3,5,6,7,8\}$, the image you
have may be the **notation** “$\{3,5,6,7,8\}$”. 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 $\{3,5,6,7,8\}$ is
the *same set* as the one denoted by $\{3,6,7,5,8\}$, but the notation $\{3,5,6,7,8\}$ is *not the same* as the notation $\{3,6,7,5,8\}$. So it is wrong to think that the notation *is*
the set.

Similarly, for real $x$, $\left\{ x|x\gt6 \right\}=\left\{ x|x\ge 6\text{ and }x\ne 6 \right\}$ even though the notation before the equals sign is different from the notationa after it.

The same set can be described by many different notations.

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:*Just like a set, 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 $A=\{3,5,6,7,8\}$ and $B$ the set of all odd integers. Then $3\in A$ and $3\in B$ and in fact $3\in A\cap B$. How can one thing be in three different containers?*Bad:*Using the sets $A$ and $B$ just defined, note that $6\in A$ and $A\in \left\{ A,B\mathrm{} \right\}$. ($\left\{ A,B\mathrm{} \right\}$ is the set whose only elements are the sets $A$ and $B$.) But $6\notin \left\{ A,B\mathrm{} \right\}$. 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
give a useful picture of relationships between a few sets portrayed as points inside a circle (or other closed curve). The circle is then a
**container** and the elements of the set are shown as points inside the circle.

The circle in the Venn diagram above represents the set $\{3,5,6,7,8\}$ shown as a subset of the integers between $1$ and $9$ inclusive. The element $3$ is shown as above $5$ and $7$ is northeast of $5$, and both those facts are completely irrelevant!

There is no good way to represent a finite set

that does not have irrelevant information

such as the location of the elements.

I prefer to think of the set $A=\{3,5,6,7,8\}$ 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 mental image is a node called $A$ that has wires or arrows connecting that node 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.)

You may see a similarity between this and a pointer in a computer language. 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.

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