Which is countable and which is uncountable ?

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In summary: If so, you can use those definitions to determine which sets are countable and which are uncountable. For (i), you can use the fact that the set of rationals is countable, and since you are working with triples of rationals, this set is also countable. For (ii), you can use the fact that the power set of a set with n elements has 2^n elements, and since the set of natural numbers is infinite, its power set is uncountable. For (iii), you can use the fact that any finite set is countable, and since the set of natural numbers is infinite, the set of finite subsets of natural numbers is also infinite and thus uncountable. In summary, the
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Homework Statement



Determine (with proofs) which of the following infinite sets are countable and which are uncountable:
(i ) The set of all triples (x, y, z) where x, y, and z are rationals;
(ii ) The set of all subsets of N;
(iii ) The set of all finite subsets of N.

Note: N is Natural Numbers

Homework Equations


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The Attempt at a Solution



For (i), There is a theorem that states all rational sets are countable, so I think it is countable is this right ? If so, I don't know how to write the correct proof.

For (ii), I think it is uncountable becasue the power set of a set S has strictly greater cardinality than S. Is this right, again I don't know how to write the proof for this one.

For (iii), I think it is countable because all sets, constituting of elements from Z (or any countable set), but where an element can occur multiple times (but only finitely many times), is also countable (so these are like subsets, except elements can occur more than once). Is this right, again I don't know how to write the proof for this one.

This is all I can do, can someone help me please ?
 
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Do you have any idea what the definitions of "countable" and "uncountable" are?
 

1. What is the difference between countable and uncountable nouns?

Countable nouns can be counted and have a singular and plural form, while uncountable nouns cannot be counted and only have a singular form.

2. How can I determine if a noun is countable or uncountable?

One way to determine if a noun is countable or uncountable is to ask if it can be counted or if it has a plural form. If the answer is yes, it is countable. If the answer is no, it is uncountable.

3. Can a noun be both countable and uncountable?

Yes, some nouns can be both countable and uncountable depending on the context. For example, "water" is uncountable when referring to the substance, but countable when referring to a specific amount or type of water.

4. Are there any rules for determining if a noun is countable or uncountable?

There are some general rules, such as most abstract nouns are uncountable, while most concrete nouns are countable. However, there are many exceptions and it is best to consult a dictionary for specific nouns.

5. How does knowing if a noun is countable or uncountable affect grammar and sentence structure?

Countable nouns can be used with articles (a/an, the) and have a plural form, while uncountable nouns do not use articles and do not have a plural form. This affects subject-verb agreement and the use of quantifiers such as "some" or "much/many".

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