Discrete Math: How can I determine the consistency of the system?

[agentlee]

Homework Statement

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 have new files, then the system software is not being upgraded."

Homework Equations

p: The system software is being upgraded
q: Users cannot access the file system
r: users can save new files.

The Attempt at a Solution

My professor did not really explain what consistency was. So I am unsure of how to go about doing this problem. I did some searching on this same problem and found that p→⇁q

I don't quite understand why that is. Can someone please help?

 

[mighty2000]

First, let's clarify what consistency means in this context. In logic, consistency refers to the relationship between statements or propositions. A set of statements is considered consistent if they can all be true at the same time. In other words, there are no contradictions or conflicts between the statements.

Now, let's break down the system specifications given in the forum post and translate them into logical statements using the given variables.

1. Whenever the system software is being upgraded, users cannot access the file system. This can be translated as p → q. In other words, if the system software is being upgraded (p), then users cannot access the file system (q).

2. If users can access the file system, then they can have new files. This can be translated as q → r. In other words, if users can access the file system (q), then they can have new files (r).

3. If the system software is not being upgraded, then users can access the file system. This can be translated as ¬p → ¬q. In other words, if the system software is not being upgraded (¬p), then users can access the file system (¬q).

Now, let's try to determine the consistency of these statements by looking for any contradictions or conflicts between them.

First, let's consider the statements p → q and q → r. If we assume that p is true (the system software is being upgraded), then q must also be true (users cannot access the file system). This makes the statement q → r automatically true, since both q and r are true. So far, there are no contradictions or conflicts between these two statements.

Next, let's consider the statements p → q and ¬p → ¬q. If we assume that p is true (the system software is being upgraded), then q must also be true (users cannot access the file system). However, the statement ¬p → ¬q states that if p is false (the system software is not being upgraded), then q must also be false (users cannot access the file system). This contradicts our initial assumption that p is true and q is true. Therefore, these two statements are inconsistent.

In conclusion, the system specifications given in the forum post are not consistent, since there is a contradiction between p → q and ¬p → ¬q. This means that the system cannot satisfy all of these specifications at the same time.