# Cardinality of Sets and Power Sets

• Sep 8th 2011, 01:22 PM
onemachine
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.
• Sep 8th 2011, 01:29 PM
onemachine
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?
• Sep 8th 2011, 01:56 PM
Plato
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 $$\emptyset$$ 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: $\displaystyle \{ \emptyset \}\in (2^A\equiv P(A))$
• Sep 8th 2011, 05:26 PM
onemachine
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