Math 2320.03

Quiz 1 Version 2 September 29,
2000

- Let f:
**Z**®**N**be defined by f(n) = |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:**The function is onto so it must have a right inverse.

Let g:**N**®**Z**be defined by g(n) = n. Then f°g = i_{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:**No. f is not one-to-one as, for example, f(1) = f(-1) and 1 ¹ -1.

- Does f have a right inverse?
- Prove that
**N**and {1,2,4,8,16,32,¼}, the set of powers of 2 in**N**, have the same cardinality by giving a formula for a bijection**N**® {1,2,4,8,16,32,¼}.

You do not need to**prove**that your answer is correct.**Answer:**f(n) = 2^{n}. - Consider the following list of
sets:
Æ,{1,3,5,7},{1,2,3,4}, {1,2,3,4,5,6,7}, **R**- 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.

{1,2,3,4,5,6,7} has cardinality 7.**R**does not have finite cardinality. - Find a set on the list with the same cardinality as
**Z**, the set of**integers**, or explain briefly why none exists.**Answer:**None exists. A set with the same cardinality as**Z**must be infinite and countable.**R**is infinite but not countable.

- 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.