9.1.6. Question 6

Try these first

1. 1.

   List three integers in the set S that are divisible by 5 but not by 7.

   Hint

   Think about multiples of 5.

2. 2.

   Is the set of all integers countable or uncountable?

   Hint

   Can you list all integers?

3. 3.

   Is the set of integers divisible by 5 but not by 7 countable?

   • True
   • False

   Hint

   Remember the process of listing integers.

4. 4.

   What is the main characteristic of the decimal representation in the discussed set of real numbers?

   • Contains only 1s
   • Contains all digits
   • Contains only 0s

   Hint

   Recall what digits are in the decimal representations.

5. 5.

   Consider a set of integers having certain divisibility conditions. Formulate a proof to show that the set of integers divisible by both 3 and 5 but excluding multiples of 10 is countable.

   Hint

   Start by outlining the integers that meet your criteria.

6. 6.

   Define a custom set involving real numbers with decimal representations consisting of 0s and 1s. Prove whether this new set is countable.

   Hint

   Look for patterns in finite binary representations.