# Solutions to Problem Set 27

Math 310-01/02

11-27-2017

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?

(a)

(b)

(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.

Define by

Note that if is odd, then is even. Further, if , then . This shows maps Y into X.

Finally,

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.)

Define by

If , then

Hence, f maps into .

Define by

If , then

Hence, maps into .

Finally,

Hence, f and are inverses. Therefore, f is bijective, and and have the same cardinality.

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