1. SUBSETS How can I send VALENTINES? --- Let me count the ways.
OF MY SET OF FRIENDS

2. S = { , , } { , , } { } { } { , } { , } { } { } { , } the POWER SET
{ , , }
{ , , }
{ }
the POWER SET
of S
{ }
{ , }
{ , }
{ }
{ }
{ , }

3. { , , }
{ , , }
{ , , }
{ , , }
{ , , }
{ , , }
{ , , }
x x

4. If a set S has n members, then
S has 2n subsets.
ie: the POWER SET of S has 2n members

5. When S is a finite set, the power set of S contains more members
Than S …….. A finite set has more subsets than members.
Cantor proved the same for infinite sets … ie: for any set S,
The power set of S has a greater cardinal number than S.
We will examine the infinite set N = { 1, 2, 3, 4, 5, ……… }
and the power set of N

6. If there is a one to one correspondence between N and P(N) it
would look like this:

8. 2 is NOT a member of S2
3 is NOT a member of S3
5 is NOT a member of S5
7 is NOT a member of S7

9. 2 is NOT a member of S2
3 is NOT a member of S3
5 is NOT a member of S5
7 is NOT a member of S7
Build a set containing all the
numbers that do NOT belong to
their paired set

10. 2 is NOT a member of S2
3 is NOT a member of S3
5 is NOT a member of S5
7 is NOT a member of S7
Build a set containing all the
numbers that do NOT belong to
their paired set … call this set
Sk and include it in the list of
subsets … it must correspond to
some natural number k.

11. Is k a member of Sk ? If it is then it isn’t. If it isn’t then it is.
= { m / m is NOT a member of Sm