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I can't understand why the following conversions are possible.

  • A = {x ∈ N : ∃n ∈ N(x = 2n)} ⋯ ①
  • = {x ∈ N : x is even} ⋯ ②

A is the set of all x is in the natural numbers, such that there is at least one natural number n such that x = 2n. In other words, there is at least n such that x is even. But ② statement means that all x is even.

So, I don't understand why the ③ is wrong, which used universal quantifiers.

  • A = {x ∈ N : ∀n ∈ ( = 2) } ⋯ ③

Below is the Textbook Citation.

A = {x ∈ N : ∃n ∈ N(x = 2n)}

Let’s look at this carefully. First, there are some new symbols to digest: “N” is the symbol usually used to denote that natural numbers, which we will take to be the set {0, 1, 2, 3, . . .}. Next, the colon, “:”, is read such that; it separates the elements that are in the set from the condition that the elements in the set must satisfy. So putting this all together, we would read the set as, “the set of all x in the natural numbers, such that there exists some n in the natural numbers for which x is twice n.” In other words, the set of all natural numbers, that are even. Here is another way to write the same set.

A = {x ∈ N : x is even}.

Asaf Karagila
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Aspas
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  • The definition of “$x$ is even” is $\exists n \in N (x = 2n)$. This is why ${x \in N \mid \exists n \in N (x = 2n)} = {x \in N \mid x \text{ is even}}$. – Mark Saving Nov 15 '22 at 02:45
  • I don't understand why the following is wrong, which used universal quantifiers instead of existential quantifiers.

    A = {x ∈ N : ∀n ∈ ( = 2) }

     – Aspas Nov 15 '22 at 07:04
  • @Aspas Can you name a natural number $x$ such that $\forall n\in\Bbb N(x=2n)$ ? – Sassatelli Giulio Nov 15 '22 at 07:05
  • @ Sassatelli Giulio Thank you. I made the mistake of thinking about the condition of n instead of x. – Aspas Nov 15 '22 at 07:19

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