SOLUTIONS SOLUTIONS

Math 2320.03
Quiz 1 Version 2     September 29, 2000

1. Let f:Z ® N be defined by f(n) = |n|.

1. 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 = iN.
2. 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.

2. 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) = 2n.
3. Consider the following list of sets:
 Æ,{1,3,5,7},{1,2,3,4}, {1,2,3,4,5,6,7}, R

1. 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.
2. 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.

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On 2 Oct 2000, 20:33.