Cardinality of Sets and Power Sets

I'm sure this is an elementary question for most of you...

Let $\displaystyle A = \{1,2, \emptyset \}$

**Determine the following:**

a) |A|

*This one is easy; It's 3, because there are 3 objects in A.*

b) |{A}|

*I believe the cardinality here is 1, but I'm not sure...can someone explain WHY?*

c) 2 ^ |A|

d) |2 ^ A |

*I think c) and d) are asking the same thing but this seems tricky to me because the empty set is an element of A. Are c) and d) equal? If so, what are they and why?*

e) Is the empty set an element of A? *YES*

f) Is the empty set a subset of A? *YES*

g) Is the set containing the empty set an element of A? *????*

h) Is the set containing the empty set a subset of A? *????*

I'm really trying to understand this so I'm looking for reasons WHY rather than just answers. Thanks for any input you can provide.

Re: Cardinality of Sets and Power Sets

Actually, I think the answer to f) is NO because the set containing the empty set is a subset of A. This means g) would be YES...? Right...or wrong?

Re: Cardinality of Sets and Power Sets

Quote:

Originally Posted by

**onemachine** **Let** $\displaystyle A = \{1,2, \emptyset \}$

c) 2 ^ |A| d) |2 ^ A |

*I think c) and d) are asking the same thing but this seems tricky to me because the empty set is an element of A. Are c) and d) equal? If so, what are they and why?*

e) Is the empty set an element of A? *YES*

f) Is the empty set a subset of A? *YES* CORRECT

g) Is the set containing the empty set an element of A? *????*

h) Is the set containing the empty set a subset of A? *????*

FYI: The LaTeX code [tex]\emptyset[/tex] gives $\displaystyle \emptyset$

For c&d) $\displaystyle \left|2^{|A|}\right|=8$

Quote:

Originally Posted by

**onemachine** I think the answer to f) is NO because the set containing the empty set is a subset of A. This means g) would be YES...? Right...or wrong?

Actually **the empty is a subset of every set**.

So in this case $\displaystyle \emptyset\in A~\&~\emptyset \subset A$.

For g) we have $\displaystyle \{ \emptyset \}\notin A$.

For h) we have $\displaystyle \{ \emptyset \} \subset A$

As an extra: $\displaystyle \{ \emptyset \}\in (2^A\equiv P(A))$

Re: Cardinality of Sets and Power Sets

I've got this all figured out. Here are the answers just in case it helps someone in the future.

**Determine the following:**

a) |A|=3

b) |{A}|=1

c) 2 ^ |A| = 8

d) |2 ^ A |= 8

e) Is the empty set an element of A? YES

f) Is the empty set a subset of A? YES

g) Is the set containing the empty set an element of A? NO

h) Is the set containing the empty set a subset of A? YES