Math 2320.03

Quiz 1 Version 1 September 29,
2000

- Let f:
**N**®**Z**be defined by f(n) = n.- Does f have a left inverse?

If no, give a proof which explains why not.

If yes, just give a formula for a left inverse of f.**Answer:**The function is one-to-one so itmust have a left inverse.

Let g:**Z**®**N**be defined by

(Note that there are many other correct choices for g.)g(n) = ì

í

în if n ³ 0 0 if n < 0

Then g°f = i_{N}. - Does f have a right inverse?

If no, give a proof which explains why not.

If yes, just give a formula for a right inverse of f.**Answer:**No. f is not onto as, for example, -1 \not Î f(**N**).

- Does f have a left inverse?
- Prove that
**N**and {0,1,4,9,16,25,¼}, the set of squares in**N**, have the same cardinality by giving a formula for a bijection**N**® {0,1,4,9,25,¼}.

You do not need to**prove**that your answer is correct.**Answer:**f(n) = n^{2}. - Consider the following list of sets:
Æ,{1,3,5,7},{1,2,3,4}, **N**,**Q**- Identify three sets on the list with the same cardinality or
explain why this is impossible.
**Answer:**Just look!

Æ has cardinality 0.

Both {1,3,5,7} and {1,2,3,4} have cardinality 4.

Neither of**N**,**Q**are finite. They do have the same cardinality À_{0}. - Find a set on the list with the same cardinality as
\mathbbR, the set of
**real numbers**, or explain briefly why none exists.**Answer:**None exists. \mathbbR is infinite as are**N**and**Q**. Both**N**and**Q**are countable but \mathbbR is not.

- Identify three sets on the list with the same cardinality or
explain why this is impossible.

File translated from T

On 2 Oct 2000, 20:33.