Question

What is the cardinality of each of the following sets?
a) {a}
b) {{a}}
c) {a, {a}
d) {a, {a}, {a, {a}}}

Answer 1

The carnality of a set is the number of unique elements in the set.
For example:
{a, b c} has three unique elements.
{a, a, b, b} has two unique elements (a and b).

But now it gets tricky. Carnality is shallow and what I mean by that is that you count elements which are sets as one element and not as the carnality of those sets.
Example:
{{a}, {b}, {c}} has a carnality of three (there are three unique elements in the first set they just happen to be new sets).

{{a, b, c}, {d, e, f}} has a carnality of two (there are two elements in the first set: {a, b, c} and {d, e, f}).

{b, c, {b}] has a carnality of three because b is not the same thing as {b} which is a set containing b.

Answer 2

That helps a lot. The answers I come up with for cardinality in the problems are:

a) 1
b) 1
c) 2
d) 4

Answer 3

@Lyrae

Answer 4

Sorry for answering late.

a), b), and c) are correct!

d) however is a bit tricky and if you look closely you'll notice set-ception (not a real mathematical term) i.e. the third element is a set containing a set and not two separate sets.

Answer 5

Oh and I just noticed my spell checker replaced all the cardinalitys with carnality in my first post, but that's obviously not correct :/