1. Suppose that X and Y are sets, and .
(a) What is ?
(b) What is ?
(c) What is the largest that can be? What is the smallest that can be?
(c) If X and Y are disjoint, then , and this is the largest that can be.
If , then , and . This is the smallest that can be.
2. Prove that the following sets of integers have the same cardinality by constructing a bijection .
Note that if is even, then is odd. Further, if , then . This shows f maps X into Y.
Note that if is odd, then is even. Further, if , then . This shows maps Y into X.
This shows that f and are inverses, so f is bijective and X and Y have the same cardinality.
3. Prove that and have the same cardinality by constructing a bijection .
Show that f is bijective by constructing the inverse function . (You must prove that f and are inverses.)
If , then
Hence, f maps into .
If , then
Hence, maps into .
Hence, f and are inverses. Therefore, f is bijective, and and have the same cardinality.
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