How do cardinalities apply to complex sets in basic set theory?

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simwun
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HI there,

Just getting into set theory just had a few questions/clarifications I guess you could call it.

if X = {Ø, a} Y={{Ø}, a} and Z = {a, {a}, {Ø}}

So i understand X has 2 elements along with Y and Z has 3.
I know what the cardinalities are of basic sets like {2, 3, 4, 5} etc but how do they apply to sets such as the above?

On top of this if U = Z - X

thats {a, {a}, {Ø}} - {Ø, a} does U = {{a}}? I am just unsure when there is a set as an element such as {Ø} do the same rules apply?

Cheers in advance,

Sim
 
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simwun said:
if X = {Ø, a} Y={{Ø}, a} and Z = {a, {a}, {Ø}}

So i understand X has 2 elements along with Y and Z has 3.
You are right about X and Z. The set Y has 2 elements: {Ø} and a.

simwun said:
On top of this if U = Z - X

thats {a, {a}, {Ø}} - {Ø, a} does U = {{a}}?
The first element a of Z is also an element of X, so it does not belong to Z - X. Next, {a} and {Ø} are not elements of X, so they belong to Z - X.

The idea is: only the "top-level" elements are real elements. E.g., {Ø} ∈ {a, {a}, {Ø}}, but Ø ∉ {a, {a}, {Ø}}.