Q6. (8 pts) Find explicit bijections to prove the following. 6.a. The set of even positive integers has the same cardinality as the set of odd positive integers. 6.b. The sets
R
and
(0,1)
have the same cardinality. [HINT: You may find the function arctan
x
useful.] 6.c. The sets
{x:EEainZ_()>1x=(1)/(a)}
and
{x:EEainZ_()>=1x=(1)/(a)}
have the same cardinality. 6.d. The sets
(0,1)
and
(0,1]
have the same cardinality. [HINT: You may find the bijection in part 6.C. useful.]

Question

Q6. (8 pts) Find explicit bijections to prove the following. 6.a. The set of even positive integers has the same cardinality as the set of odd positive integers. 6.b. The sets

R

and

(0,1)

have the same cardinality. [HINT: You may find the function arctan

x

useful.] 6.c. The sets

{x:EEainZ_()>1x=(1)/(a)}

and

{x:EEainZ_()>=1x=(1)/(a)}

have the same cardinality. 6.d. The sets

(0,1)

and

(0,1]

have the same cardinality. [HINT: You may find the bijection in part 6.C. useful.]

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