Let S = Ø. Is the power set of S, P(S) = {Ø} ?
It is not correct to say P(S) = Ø, right?
Also, is it true to say {Ø} ⊆ Ø ?
What I'm getting at is: if S = Ø, then is the power set of S a subset of S ?
Thanks.
Yes, assuming "P(S)" means the "power set" of S, the set of all subsets of S, P(S)= P(Ø)= {Ø}
Right. One important difference is that P(S)= P(Ø) contains a single object, Ø itself, so is not empty.It is not correct to say P(S) = Ø, right?
No, what is true is the other way around Ø ⊆ {Ø}. Since {Ø} has one member and Ø has none {Ø} cannot be a subset of Ø.Also, is it true to say {Ø} ⊆ Ø ?
No, it is not. S is a subset of the power set of S (that's true for any set S).What I'm getting at is: if S = Ø, then is the power set of S a subset of S ?
Thanks.
It's often instructive to look at "small examples" (often called "toy examples" by professionals in the field).
So let's create a small set:
S = {Alice, Bob}.
The power set can be thought of as "teams we can make" from this set. Let's look at it now:
P(S) = {Ø,{Alice}, {Bob}, {Alice,Bob}}
Note there is a difference between "the team with only Alice on it" and "Alice"; Alice is a member of her one-woman team, but Alice is not a "member" of herself.
The difference between Ø and {Ø}, is the same kind of difference between an empty bag, and a bag with an empty bag INSIDE it (which is thus not empty, even though the bag inside the bag doesn't have any stuff in it).
In computer theory, one would say that these objects are of different TYPES: Ø is a null instruction (like an blank line in a program which can be safely deleted), while {Ø} is a reference to a null instruction (which is thus not null, it requires code).
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Our usual way of describing "containment" in English doesn't help much here: when we say "A is in B", do we mean: "A is one of the things which we call collectively, B" (Tom in is my grade at school), or do we mean that "B is bigger than A" (the living room is in the house)? It's easy to become confused in set theory: the elements of a set are usually "other sets", and we often have "many layers" to drill down to "atomic elements" (and strictly speaking, the only "atomic element" we are absolutely certain of in Zermelo-Fraenkel set theory is the empty set and thus (via the axiom of infinity) its minimal closure under "sucessorship" (for a set A its successor is A U {A})).
For example, one cannot prove that my "toy set" above (with Alice and Bob in it), IS actually a set. Set theory doesn't actually SAY what things might be "set elements" except for Ø, but using this, we can construct natural numbers, rational numbers, etc., all the way up to many kinds of things, but we cannot, in general, use set theory to reason about "everyday objects" unless we posit the existence of "a universe of discourse". Such a hypothesis is typically taken to be a starting point of a logicial system: we take as given "stuff we want to talk about". Formally, we might start with an alphabet: {A,B}, and manipulate this alphabet purely formally, thinking of the MODEL: A = Alice, B = Bob.
One has to be careful, though. Sets are an ABSTRACTION, which follow certain "arbitrary" rules we HOPE reflect the way we reason about things, but Alice and Bob may, in fact, be real people. One may use arithmetic to balance your check-book, but the goods represented by the purchasing power of the currency represented by the numbers, are not "elements of arithmetic". In practice, there's not too much to worry about, the current rules of set theory don't seem to lead to any paradoxical results (math seems to "work"), but try to keep in the back of your mind there is a certain limitation to when we say: "X is TRUE" in math, we don't mean this is an actual fact, it is conditional under certain assumptions (hopefully, reasonable ones).
Overall the process is something like:
Real world--->abstraction to formal system (leaves out non-essential information)
Formal manipulation in formal system (very strict rules here!)
Result--->possible application in real world (only "possible" because we can never know for SURE what information is truly "essential").
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One thing you should take to heart early, if you want to avoid heartache in set theory is: A and {A} are NEVER the same set, they live on different "levels".