Negating statements with quantifiers in them

Question

First statement,

∀ odd integers n, ∃ an integer k such that n = 2k + 1

Second statement,

∃ m ∈ ℝ such that ∀ n ∈ ℝ, m · n = n

Before the negation, I'd like to ask tips on how to translate this into English, removing the symbols and variables.

This is what I think.

For the first one: For all of the odd integers, there exists an integer such that the odd integer is equal to two times the integer plus one.

Second one: There exists at least one number in the set of real numbers such that for each number in the set of real numbers, the first number times the second number is equal to the first.

Any suggestions?

For the negation..well any help would be appreciated.

Answer 1

$a)$: For any odd integer $n$, there is some integer $k$ such that: $n = 2k+1$,

$b)$: There is a real number $m$ such that for any real number $n$: $m\times n = n$.

Answer 2

Dual Negation: $$\neg \forall x\in S : P(x) \;\iff\; \exists x \in S:\neg P(x) \\\neg \exists x\in S : P(x) \;\iff\; \forall x \in S:\neg P(x)$$

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When in doubt, take it one step at a time.

So "not $\forall$ odd integers n, $\exists$ integer $k$ such that $n=2k+1$"

Becomes "$\exists$ odd integers n, not $\exists$ integer $k$, such that $n=2k+1$"

Then "$\exists$ odd integers n, $\forall$ integer $k$, such that not $n=2k+1$"

Finally "$\exists$ odd integers n, $\forall$ integer $k$, such that $n\neq 2k+1$"

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Can you complete the other question now?