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1. All natural numbers are Integer.

So I know natural number is N and integer is Z but how do I translate this statement into Symbolic Form ?

user832845
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1 Answers1

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$$\forall x\in\mathbb N\space\space\space x\in \mathbb Z$$

Alessio K
  • 10,599
  • It is either $\forall x~(x\in\Bbb N\to x\in\Bbb Z)$ xor $\forall x{\in}\Bbb N~x{\in}\Bbb Z$. Do not mix and match. – Graham Kemp Oct 06 '20 at 02:36
  • But what's wrong with the "$\implies$" above? I read the above as "For all natural numbers x, it implies that $x$ is an integer", that is all natural numbers are integers. – Alessio K Oct 06 '20 at 07:07
  • An implication requires an antecedant and a consequent; both of which are well formed formula on there own. The quantifier binding of a variable cannot act as an antecedant as it is not in itself a well formed formula. – Graham Kemp Oct 06 '20 at 22:47
  • Likewise, a well formed quantified statement has a variable binding and a scope which must contain a well formed formula. A statement that begins with an implication symbol, such as: ${\implies}x{\in}\Bbb Z$ , is not a well formed formula. @Äres – Graham Kemp Oct 06 '20 at 22:52
  • A formula beginning with a quantifier is called a quantified formula. The statement above does not begin with $\implies x\in\mathbb Z$. The statement $ \forall x \in A \Rightarrow x \in B $ means that every element in $ A $ is also an element of $ B.$ Anyway, I've fixed my answer. Thanks. – Alessio K Oct 07 '20 at 07:37
  • A quantified statement has two components: the variable declaration, and the scope over which it is bound. The scope comes after the declaration and must be a well formed formula. Your declaration is $\forall x\in A$ and thus the scope contains $\implies x\in B$ and is thus not well formed. – Graham Kemp Oct 07 '20 at 11:20