Express
$$\forall\ n\in\mathbb N\ \exists\ m \in\mathbb N: \ n^4 = m^2$$
in words without using the symbol $\mathbb N$.
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4 Answers
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To avoid usage of the term "perfect power" (in case you didn't know it), you can say
Every fourth power of a natural number is also the square of a natural number.
AlexR
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Assuming $n\in \mathbb{N}$, this can be spoken as:
"For all natural numbers there exists a natural number such that the fourth power of the first is the square of the second." i.e. "For all natural numbers every fourth power is a perfect square."
dos
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For every n belongs to natural numbers there exists a m belongs natural numbers such that fourth power of n is equal to square of m.
Jasser
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1Though correct, this is merely a translation of the notation into incomplete english. – AlexR Sep 20 '14 at 10:29
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I was assuming that he also don't know the meaning of the symbols @AlexR – Jasser Sep 20 '14 at 10:31
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1Agreed with AlexR. Especially when there are lots of quantifiers, the hard part is translating the sentence into natural English, not just mechanically replacing $\forall $ with "for all" and $\exists$ with "there exists". – beep-boop Sep 20 '14 at 10:43
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@user291957 The question sounds more like an assignment to me, i.e. the OP is specifically asked to translate the expression. This doesn't imply the OP wasn't able to understand the quantifiers. – AlexR Sep 21 '14 at 16:41