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“100 is the maximum natural number that satisfies P.” what is each translation saying?

P(100) ∧ ∀y ∈ N, P(y) ⇒ y ≤ 100 vs.

P(100) ∧ (∀y ∈ N, P(y) ⇒ y ≤ 100)

and then simply

∀y ∈ N, P(y) ⇒ y ≤ 100

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    Each translation is saying “100 is the maximum natural number that satisfies P.”. – 2017-02-08
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    omitting the p(100) at the beginning in the last does nothing? – 2017-02-08
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    Well, $P(100)$ indicates that $100$ necessarily satisfies $P$. – 2017-02-08
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    It might be wise to add brackets to your expressions to signify the order of operations you are using. For instance the $\rightarrow$ could be read in many different ways without brackets. – 2017-02-08

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The first two statements differ only by parenthesis used to clarify order of operations.   However, I'd suggest that enclosing the implication is the proper form, as you want to clearly bind the entire implication to the universal quantifier.

$P(100)~\wedge~\forall y\in \Bbb N ~( P(y)\to y\leq 100)$ says that "$100$ satisfies $P$, and if any natural number satisfies $P$, then it is not greater than $100$."

$\forall y\in \Bbb N ~( P(y)\to y\leq 100)$ just says "If any natural number does satisfy $P$ then it is not greater than $100$."   It in no way ensures that any numbers do satisfy $P$; it just identifies numbers that do not.

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    ohhh thanks for this =) – 2017-02-09