problem from Kenneth Rosen's Discrete Mathematics and its Applications, Section 1.5 - 1.7

Question

Please solve the question. 

Question: Express each of these system specifications using predicates, quantifiers, and logical
connectives.

a. Any user with a Gmail account can access services from any Google products.

b. There is a database which contains the records of all the students in this university.

c. No servers can be down simultaneously during a power failure.

d. There is a node whose adjacent nodes are not connected to each other.
 

Answer 1

A. Watch it carefully, any means here for all

G(x) : User has gmail account                  S(x) : Services from google products 

We can frame the statement also as : if user has google account then he can access services google products, or having a google account is sufficient for accessing services of google products

 $P(x) : \forall_{x} (G(x) \rightarrow S(x)))$

 

B. Watch it carefully, there means here there exists

D(x) : Database                           R(x): Records of all students

The crux of the question is if there is a database records must also exist, or if database occurs records must be present

$P(x) : \exists _{x}(D(x)\wedge R(x))$

 

C. This can also be framed as NOT(there exists servers which is down during power failure)

S(x) : Server                              P(x) :Servers has power

$P(x) : \rightharpoondown\exists _{x}(S(x)\wedge \rightharpoondown P(x))  = \forall_{x} (S(x) \rightarrow P(x)))$

 

D. Watch it carefully, there means here there exists

N(x) :Node                                  C(x): Node x is connected to its adjacent nodes  

We may also frame the question as NOT(Every node is connected to its adjacent nodes)

$\rightharpoondown\forall_{x}(N(x)\rightarrow C(x)) = \exists (N(x) \wedge \rightharpoondown C(x))$

 

Hope this helps :)

Corrections are appreciated