SUBSETS How can I send VALENTINES? --- Let me count the ways.

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Presentation transcript:

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

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

{ , , } { , , } { , , } { , , } { , , } { , , } { , , } 2 x 2 x 2

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

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

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

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

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

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.

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