Yes, integers not divisible by 3 are countable.
The set of integers not divisible by 3 includes numbers like 1, 2, 4, 5, 7, 8, and so on. A number is not divisible by 3 if it leaves a remainder of 1 or 2 when divided by 3. We can represent these integers as either 3k + 1 or 3k + 2, where k is an integer.
Mapping to Positive Integers
To show that this set is countable, we need to demonstrate that we can create a one-to-one correspondence (a bijection) between the integers not divisible by 3 and the set of positive integers. Here’s how:
- We can establish a mapping where:
- 1 maps to 1
- 2 maps to 2
- 4 maps to 3
- 5 maps to 4
- 7 maps to 5
- 8 maps to 6
- and so on...
- This mapping covers all integers not divisible by 3 without any omissions, and every positive integer will have a unique counterpart in the set of integers not divisible by 3.
Formal Explanation
More formally, we can define a function *f* that maps positive integers to the integers not divisible by 3. For every positive integer *n*:
- If n is odd, we assign *f(n) = (n+1)/2 * 3 - 2*
- If n is even, we assign *f(n) = n/2 * 3 - 1*
Conclusion
Because we have demonstrated a way to list out and pair the integers not divisible by 3 with the positive integers, we can conclude that the set of integers not divisible by 3 is countable, as specified in the reference. This means that even though there are an infinite number of integers not divisible by 3, they can be arranged in a list and can be counted using a mapping to the positive integers.
