No, not all infinite sets are countable.
Understanding Countability
Countability refers to whether the elements of a set can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, ...). A set is considered countable if it is either finite or countably infinite. The reference highlights the distinction: when we want to emphasize that we're talking about infinite sets, we use the term "countably infinite." This avoids confusion with finite sets, which are also countable.
Countably Infinite vs. Uncountable
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Countably Infinite: An infinite set is countably infinite if its elements can be listed in a sequence, meaning you can count them, even though the counting never ends. The set of integers (..., -2, -1, 0, 1, 2, ...) is countably infinite because you can create a way to list all integers.
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Uncountable: An infinite set is uncountable if it's "too large" to be put into a one-to-one correspondence with the natural numbers. This means no matter how you try to list its elements, you'll always miss some. The set of real numbers between 0 and 1 is a classic example of an uncountable set.
Examples
| Set | Countable? | Explanation |
|---|---|---|
| Natural Numbers (N) | Yes | By definition, they are the basis for counting. |
| Integers (Z) | Yes | Can be arranged in a list: 0, 1, -1, 2, -2, 3, -3, ... |
| Rational Numbers (Q) | Yes | Although they seem dense, they can be systematically listed. |
| Real Numbers (R) | No | Proven by Cantor's diagonalization argument. There are "more" real numbers than natural numbers. |
| Real Numbers between 0 and 1 | No | A subset of the Real numbers, and therefore also uncountable |
Conclusion
Infinite sets can be either countable (countably infinite) or uncountable. The key distinction lies in whether the elements of the set can be put into a one-to-one correspondence with the natural numbers. If yes, it's countable; if not, it's uncountable.
