Logic basic

Question

Express each of these system specification using predicates , quantifier and logical connectives if necessary

1) At least one console must be accessible during every fault condition

2) The email address of every user can be retrived whenever the archieve contain atleast one message sent by user on the system

3) For every security breach there is atleast one mechanism that can detect that breach if and only if there is a process that has not been compromised

4) There are atleast 2 paths connecting every 2 distinct endpoints on the network

5) No one knows the password of every user on the system except system administrator who knows all passwords

Answer 1

Solution Please point out my mistakes and help for Q5 is needed :)

1)At least one console must be accessible during every fault condition

∀x∃yA(x,y)

where A(x,y) mean Fault condition x must access y console

2) The email address of every user can be retrived whenever the archieve contain atleast one message sent by every user on the system

=∀u∃m(Archieve(m,u)--->∀uemailid( u ))

where Archieve(m,u) contain mail m sent by user u

emailid( u )- email id of user u can be retrived

3) For every security breach there is atleast one mechanism that can detect that breach if and only if there is a process that has not been compromised

∀p∃sDetect(p,s)<---> ∃x∼Compromised(x)

where detect(p,s) mean Security breach p has one mechanism s That help us to detect breach

Compromised(x) : process x is compromised

4) There are atleast 2 paths connecting every 2 distinct endpoints on the network

∃p1∃p2((p1!=p2)⋀ ∀x∀y((x!=y) ⋀ Connect (p1,x) ⋀ Connect (p2,y)⋀ ∼Connect (p1,y) ⋀ ∼Connect (p2,x))

where connect (pi,x) mean Path p1 is connected to vertex x

5) No one knows the password of every user on the system except system administrator who knows all passwords

// not getting i am able to write just alf part for system adimin help here please

Comments

5)∀x(SystemAdmin(y)->password(x)) V ∀x ~(Someone(y)-> password(x))