Algebra Know-It-ALL

elements,” and you’ve named all the elements of the null set. When a set is infinite, you can’t
name all of its elements, no matter how much time you have.
Even if a set is infinite, you might be able to write an “implied list” that reveals exactly
what all of its elements are. Consider this:

W= {0, 1, 2, 3, 4, 5, ...}

This is the set of whole numbers as it is usually defined in mathematics. You know whether or
not something is an element of set W, even if it is not shown above, and even if you could not
reach it if you started to scribble down the list right now and kept at it for days. You can tell
right away which of the following numbers are elements of W, and which are not:

12

1/2

23

100/3

78,883,505

356.75

90,120,801,000,000,000

−65,457,333

The first, third, fifth, and seventh numbers are elements of W, but the second, fourth, sixth,
and eighth numbers are not.
Some infinite sets cannot be totally defined by means of any list, even an “implied list”!
You’ll learn about this type of set in Chap. 9.

Sets within sets

A set can be an element of another set. Remember again, anything can be a member of a set!
You can have sets that get confusing when listed. Here are some examples, in increasing order
of strangeness:

{1, 2, 3, 4, 5}

{1, 2, {3, 4, 5}}

{1, {2, {3 ,4, 5}}}

{1, {2, {3, {4, 5}}}}

{1, {2, {3, {4, {5}}}}}

An “inner” or “member” set can sometimes have more elements than the set to which it
belongs. Here is an example:

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

Here, the main set has three elements, one of which is a set with six elements.

The Concept of a Set 21