Question

Are these system specifications consistent? “Whenever the system software is being upgraded, users cannot access the file system. If users can access the file system, then they can save new files. If users cannot save new files, then the system software is not being upgraded.”

Short answer

From the given statements,the system is consistent.

Step-by-step solution

Introduction to the Concept

"If-then" is a logical sign. It can be depict as $\mathbf{\to }$.

"not"is a logical sign. It can be depict as $\mathbf{\bigsqcup }$.

Given statements

The given statements are,

P = The software is upgraded

Q = users can access the file system

R = users can save new files

Solution Explanation

The system specifications are translated as given below,

P→~Q

Q→R

~R→~P

For the system specifications, create a truth table.

Assume that P,Q, & Rhave all conceivable values.

P	Q	R
T	T	T
T	T	F
T	F	T

T	F	F
F	T	F
F	T	T
F	F	T
F	F	F

The Truth table of P→~Q, Q→R, and ~R→~P is shown below

P	Q	R	~P	~Q	~R	P→~Q	Q→R	~R→~P
F	F	F	T	T	T	T	T	T
F	F	T	T	T	F	T	T	T
F	T	F	T	F	T	T	F	T
T	F	F	F	T	T	T	T	F
F	T	T	T	F	F	T	T	T
T	F	T	F	T	F	T	T	T
T	T	F	F	F	T	F	F	F
T	T	T	F	F	F	F	T	T

Now, using the notion of negation, find the negation of P, Q, and R.

P	Q	R	~P	~Q	~R
T	T	T	F	F	F
F	F	F	T	T	T
T	T	F	F	F	T
T	F	T	F	T	F
T	F	F	F	T	T
F	T	T	T	F	F
F	T	F	T	F	T
F	F	F	T	T	F

It can be seen from the table that there are four rows that make all of the compound propositions true.

All of the specifications are true for the consistent system states.

P	Q	R
T	F	T
F	T	T
F	F	T
F	F	F

Thus the system is consistent.