Convert the following statement to predicate logic.
At least one router is functioning normally if the throughput is between 100kbps and 500 kbps and the proxy server is not in diagnostic mode.
Solution:
Let the domain be all applications and routers. Let Router(x) be “x is a router”, and let ProxyServer(x) be “x is the proxy server.” Let Diagnostic(x) be “x is in a diagnostic state”. Let ThroughputNormal be “the throughput is between 100kbps and 500 kbps”. Let Functioning(y) be “y is functioning normally”.
∀x (ThroughputNormal ∧(ProxyServer(x)∧ ¬Diagnostic(x))) → (∃y Router(y)∧Functioning(y))
I think the answer should be
∀x (ThroughputNormal ∧(ProxyServer(x)∧ ¬Diagnostic(x)) → (∃y Router(y)∧Functioning(y)))
I am having doubt regarding that paranthesis in bold.
Recall $\forall x~(P(x)\to Q) \iff (\exists x~P(x))\to Q\\ (\forall x~P(x))\to Q\iff \exists x~(P(x)\to Q)$
This has the form of $\forall x~(T\wedge P(x))\to Q$. The order of operations means this is implicitly: $(\forall x~(T\wedge P(x)))\to Q$, and is equivalent to $T~\to~\exists x~\big(P(x)\to Q\big)$
"If throughput is normal then there is a proxy server that if it is not in diagnostic mode, then there is a functioning router." or "There is a proxy server such that, some router will function normally if throughput is normal and that proxy server is not in diagnostic mode."
This is not quite what you want to say.
This has the form of $\forall x~\big(T\wedge P(x)~\to~Q\big)$, which is equivalent to $T~\to~\forall x~(P(x)\to Q)$
"Whenever throughput is normal, any proxyserver that is not in diagnostic mode guarantees that there is a functioning router," or "There is a functioning router if throughput is normal and any proxyserver is not in diagnostic mode."
So, this is closer to what you want to say.
How many Proxy Servers are there?