Question on Predicates and Quantifiers I am reading from "Discrete Mathematics and Its applications" by Kenneth H. Rosen, 7th edition. Consider the highlighted part in the following example taken from the same book:


  Question Use predicates and quantifiers to express the system specifications “Every mail message larger
  than one megabyte will be compressed” and “If a user is active, at least one network link will
  be available.”
  
  Solution: Let S(m, y) be “Mail message m is larger than y megabytes,” where the variable x has
  the domain of all mail messages and the variable y is a positive real number, and let C(m) denote
  “Mail message m will be compressed.” Then the specification “Every mail message larger than
  one megabyte will be compressed” can be represented as ∀m(S(m, 1) → C(m)).
  Remember the rules of
  precedence for quantifiers
  and logical connectives!
  Let A(u) represent “User u is active,” where the variable u has the domain of all users,
  let S(n, x) denote “Network link n is in state x,” where n has the domain of all network
  links and x has the domain of all possible states for a network link. Then the specification
  “If a user is active, at least one network link will be available” can be represented by
  
  ∃uA(u) → ∃nS(n, available).


"Existential quantifier" ∃u is used here, So I think, it should be ∃uA(u)∧ ∃nS(n, available) in place of ∃uA(u) → ∃nS(n, available).

Generally, we use 'implication' with 'universal quantifier' as shown in ∀m(S(m, 1) → C(m)). 

Please correct me, if I am wrong.
The meaning of
$$ \exists u A(u) \land \exists n S(n,\text{available}) $$
is "there exists an active user, and there exists a network link which is available". The conditioning is missing here.

If $P = \exists u A(u)$ and $Q = \exists n S(n,\text{available})$, then we are trying to model "if $P$ then $Q$", whose formal form is $P \rightarrow Q$, whereas your answer is $P \land Q$.

In particular, if there doesn't exist an active user, then $P \land Q$ is always false, whereas $P \to Q$ is always true.