Problem 5E

Show that a finite group of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.

Step 1 :

In the given problem we have to find given sets are countable or uncountable.

(a) integers not divisible by 3.

Ans ; These are countable sets, starting with number of alternative positive and negative,

counting each integer and skipping the multiples of 3.

This gives the list

11

2-1

32

4-2

53

6-3

.

.

.etc

.’. Integers not divisible by 3 is countable set.

Step 2 ;

(b) integers divisible by 5 but not by 7.

Ans : these are countable sets, we starting with 5 and alternative positive to negative counting each multiple of 5 and skipping the multiples of 7.

This gives the list

15

2-5

310

4-10

515

6-15

.

.

.etc

.’. integers divisible by 5 but not by 7 countable set.