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Question Number 3551 by prakash jain last updated on 15/Dec/15
Prove that P set of prime numbers is countable.
$$\mathrm{Prove}\:\mathrm{that}\:\mathbb{P}\:\mathrm{set}\:\mathrm{of}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{is} \\ $$$$\mathrm{countable}. \\ $$
Commented by Filup last updated on 15/Dec/15
ah, i understand the question now. I can′t answer it, though
$$\mathrm{ah},\:\mathrm{i}\:\mathrm{understand}\:\mathrm{the}\:\mathrm{question}\:\mathrm{now}. \\ $$$$\mathrm{I}\:\mathrm{can}'\mathrm{t}\:\mathrm{answer}\:\mathrm{it},\:\mathrm{though} \\ $$
Commented by Filup last updated on 15/Dec/15
If by countable, you mean there is a finite number of primes, then you are incorrect. Reason is: all numbers have prime factors let n∈Z ∴n=p_1 ×...×p_k prime factors p_i ∈P n+1≠n ∴different prime factors. all primes are prime factors of a number beacuse p_1 ×...×p_k =n a single prime is the prime factor of an infinite amount of numbers
$$\mathrm{If}\:\mathrm{by}\:\mathrm{countable},\:\mathrm{you}\:\mathrm{mean}\:\mathrm{there} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{finite}\:\mathrm{number}\:\mathrm{of}\:\mathrm{primes},\:{then}\:{you} \\ $$$${are}\:{incorrect}. \\ $$$$ \\ $$$$\mathrm{Reason}\:\mathrm{is}: \\ $$$${all}\:{numbers}\:{have}\:{prime}\:{factors} \\ $$$$\mathrm{let}\:{n}\in\mathbb{Z} \\ $$$$\therefore{n}={p}_{\mathrm{1}} ×…×{p}_{{k}} \:\:\:{prime}\:{factors} \\ $$$${p}_{{i}} \in\mathbb{P} \\ $$$$ \\ $$$${n}+\mathrm{1}\neq{n}\:\:\:\:\:\therefore{different}\:\mathrm{prime}\:\mathrm{factors}. \\ $$$$ \\ $$$$\mathrm{all}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{prime}\:\mathrm{factors}\:\mathrm{of}\:\mathrm{a}\:\mathrm{number} \\ $$$$\mathrm{beacuse}\:{p}_{\mathrm{1}} ×…×{p}_{{k}} ={n} \\ $$$$ \\ $$$$\mathrm{a}\:\mathrm{single}\:\mathrm{prime}\:\mathrm{is}\:\mathrm{the}\:\mathrm{prime}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{infinite}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{numbers} \\ $$
Commented by prakash jain last updated on 15/Dec/15
A countable set may be finite or infinite. Essentially it means that every element of set can be associated with natural number.
$$\mathrm{A}\:\mathrm{countable}\:\mathrm{set}\:\mathrm{may}\:\mathrm{be}\:\mathrm{finite}\:\mathrm{or}\:\mathrm{infinite}. \\ $$$$\mathrm{Essentially}\:\mathrm{it}\:\mathrm{means}\:\mathrm{that}\:\mathrm{every}\:\mathrm{element} \\ $$$$\mathrm{of}\:\mathrm{set}\:\mathrm{can}\:\mathrm{be}\:\mathrm{associated}\:\mathrm{with}\:\mathrm{natural}\:\mathrm{number}. \\ $$$$ \\ $$
Commented by prakash jain last updated on 15/Dec/15
P⊂N, since N is countable P is countable.
$$\mathbb{P}\subset\mathbb{N},\:\mathrm{since}\:\mathbb{N}\:\mathrm{is}\:\mathrm{countable}\:\mathbb{P}\:\mathrm{is}\:\mathrm{countable}. \\ $$

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