Let S = {x | x = pa, a E Z, (for all) p E Z+} "a" is a particular integer that is specified.

• (a) For infinite values of a the cardinality of S is infinite.
  
• (b) For infinite values of a the cardinality of S is finite.
  
• (c) For finite values of a the cardinality is finite.
  
• (d) For no value of a the cardinality of S is finite.
  
• (e) For only one particular value of a the cardinality of S is finite.
  
• (f) a and b.
  
• (g) b and c
  
• (h) a and c
  
• (i) a c and e
  
• (j) For finite values of a the cardinality of S is infinite and (c).
  

Why is i the answer. More specifically, why is c a valid choice?

When the answer says "infinite/finite values" it means for "number of a's". Not that the value itself is infinite or finite. Apparently there was a lot of confusion about this in class.

Why I don't think c is correct:
For finite values of a the cardinality of S is finite.
My issue with this is that if we consider a to only be 1 value, then the cardinality is infinite as x is composed of a * p where p is all Z+. So if that value was 1, S = {1, 2, 3, 4, 5....}. The only way to get a finite cardinality is if a is 0.
I ended up getting this one right because a and e are definitely the answer and i was the only one that had a and e as a choice.

Maybe instead of "if a takes on finitely many values then S has finite cardinality" they mean "there are finitely many values of a that will give S finite cardinality"

I could be wrong though

This post was edited by Obstacle1 on Nov 12 2019 11:28am

I don't get the difference. a taking on finitely many values and there are finitely many values of a mean the same thing?

That's not what I said. Reread what I put in quotes. Their meanings are not the same

To clarify:

It is true that "there are finitely many values of a that will give S finite cardinality" because the set of all values of a that gives S finite cardinality only contains one value which is 0

It is not true that "if a takes on finitely many values, then S will have finite cardinality" because if a takes on a single value other than 0, then S will not have finite cardinality

Do you see the difference now? I think your teacher meant for the statement to be the first statement I wrote in this post

This post was edited by Obstacle1 on Nov 19 2019 05:29pm

I came here for math, im leaving with no dignity.

This post was edited by Darkholder03 on Nov 19 2019 07:32pm

Okay, yeah that makes a lot more sense now. Thanks for the clarification.