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Consider the statemnt: $∀x∈\mathbb{N}, ∃y∈\mathbb{N}:y >(x−1)∧ Composite(x) $ .Does this translate to there are infinitely many numbers which are 1 less than a composite number? Or what is the translation?

thanks

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    The formula is clearly wrong. The part $\forall x \in \mathbb N \ ( \text {Composite}(x) \ldots )$ reads : "every number is *composite*". – 2017-01-25
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    Maybe : $∀x ∈ \mathbb N \ \exists y \in \mathbb N \ (\text {Composite}(x) \to y > (x-1))$. – 2017-01-25
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    But it makes little sense... for every natural number $n$ (not zero !) there is a number $y$ such that : $y > (n-1)$. – 2017-01-25
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    Logic is not about translating mathematical statements into (difficult to read) symbols... it is about using symbols to write formally and precisely mathematical statement, in order to avoid mistakes. – 2017-01-25
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    that is how to show there are infinitely many numbers right...ie for every number there is some number greater? – 2017-01-25

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The logic expression

$∀x∈\mathbb{N}, ∃y∈\mathbb{N}:y >(x−1)∧ Composite(x) $

does not translate to your English expression

'there are infinitely many numbers which are 1 less than a composite number',

but rather, it translates to:

'For every natural number $x$, there is a natural number $y$ greater than $x-1$, and $x$ is composite'

... which is a 'weird' sentence, since

1) it is not true that every natural number is composite

2) Assuming the domain is the natural numbers, there may be an issue with $x-1$ (what if $x$ is $0$?)

3) For any $x>0$, of course there exists some $y$ greater than $x -1$ .... $x$ itself!

(also: I disregarded the fact that the expression really needs parentheses, or else the $x$ in $Composite(x)$ is not quantified, and this is not even a proper sentence in the first place!)

All this leads me to suspect that you created this logic sentence, and that your prompt was to translate the English sentence 'there are infinitely many numbers which are 1 less than a composite number' ... which itself is indeed a much more interesting (and true) claim. Am I right?

So: let's see how to translate that English sentence into logic:

First, one way to say that there are infinitely many of such numbers is to say that

1) there is at least one such number

and

2) For every such number there is a greater such number

Now, you used a predicate $Composite(x)$, but since you are already using mathematical operators like $>$ and $-$ in your formula, let's express that '$x$ is a composite number by' by expressing that 'x has a divisor other than 1 and itself', and for that we can use the formula:

$\exists y \exists z (x = y * z \land y \not = 1 \land y \not = x)$

Then, to say that '$w$ is 1 less than a composite number $x$', we can use:

$\exists x (\exists y \exists z (x = y * z \land y \not = 1 \land y \not = x) \land w = x - 1)$

OK, so again, we need at least one of these numbers, and that for each one of these number, we can always finds a bigger one of these numbers:

$\exists w \exists x (\exists y \exists z (x = y * z \land y \not = 1 \land y \not = x) \land w = x - 1) \land \forall w (\exists x (\exists y \exists z (x = y * z \land y \not = 1 \land y \not = x) \land w = x - 1) \rightarrow \exists w' (\exists x (\exists y \exists z (x = y * z \land y \not = 1 \land y \not = x) \land w' = x - 1) \land w < w')$