Section 10.4, Due on Monday March 17th

Section 10.4 - Comparing Cardinalities of Sets

Two nonempty sets A and B have the same cardinality if there exists a bijective function f : A -> B.

Theorems:

• For every nonempty set A, the sets P(A) and 2^A are numerically equivalent.
• If A is a set, then |A| < |P(A)|.

What was difficult for me?

I learned the definition of the cardinality of a set earlier this semester.  It was a foreign concept to me, but know I understand it.  The definition, for those of you that don't know, is below.

The understanding of the cardinality is not too difficult, but the proving of the difference of cardinalities of sets has been difficult for me to understand.  I understand that you have to show a bijective function, which is simple enough.  It's the in between steps (explaining, next steps, etc.) that are tough for me.  Those are the parts that I mess up on and I will need more practice before I master these concepts.

What did I find enjoyable?

I like the enjoyable nature of cardinality - it's counting!  I like counting.  When I was a kid I would count things.  

I would take 1+1=2. 

Then 2+2=4.  

Then 4+4=8.  

8+8=16.  

16+16=32.  

32+32=64.  

64+64=128.  

128+128=256.  

256+256=512.  

512+512=1024.  

1024+1024=2048.

I JUST DID THAT FROM MEMORY. That's what I find interesting about cardinality and counting.  Now it will be fascinating to learn more about comparing cardinality.  It can't be that much harder than counting, right?

And, I also think it's interesting that cardinality and cardinals have nothing to do with each other.