How do you express system specifications using logical propositions p, q, and r?

[VinnyCee]

Homework Statement

Express the system specifications using the propositions p "The user enters a valid password," q "Access is granted," and r "The user has paid the subscription fee" and logical connectives.

a) The user has paid the subscription fee, but does not enter a valid password.

b) Access is granted whenever the user has paid the subscription fee and enters a valid password.

c) Access is denied if the user has not paid the subscription fee.

d) If the user has not entered a valid password but has paid the subscription fee, then access is granted.

Homework Equations

p = "The user enters a valid password"
q = "Access is granted"
r = "The user has paid the subscription fee"

The Attempt at a Solution

a) $$r\,\wedge\,\neg\,p$$

b) $$q\,\longleftrightarrow\,(r\,\wedge\,q)$$

c) $$\neg\,r\,\longrightarrow\,\neg\,q$$

d) $$(\neg\,p\,\wedge\,r)\,\longrightarrow\,q$$Do these answers look right?

 

[radou]

Everything looks right, except you made a typo (I guess) in b), it should be $$q\,\longleftrightarrow\,(r\,\wedge\,p)$$, unless I'm missing something.

 

[matt grime]

Even modulo typos, b is not right. The implication is not both ways. It does not assert that the only way to get access is with subscription and a valid password.

 

[radou]

Even modulo typos, b is not right. The implication is not both ways. It does not assert that the only way to get access is with subscription and a valid password.

Excuse my mistake, it should be $$(r \wedge p) \Rightarrow q$$.

 

[VinnyCee]

So they are all right except that B is really $$(r\,\wedge\,p)\,\longrightarrow\,q$$?

 

[radou]

So they are all right except that B is really $$(r\,\wedge\,p)\,\longrightarrow\,q$$?

Yes, you already have two answers which implied that.