What is your answer to this Logic question? (Mine included)

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The discussion revolves around the conditions under which a member of a frequent-flyer club may have their membership terminated. Participants analyze the logical implications of various conditions, particularly focusing on whether failing to earn mileage in the first twelve months is sufficient for termination. There is debate on how to structure the logical expression representing these conditions, with differing opinions on the sufficiency of condition p compared to others. Ultimately, the conversation highlights the complexity of the rules and the need for careful interpretation of exceptions. The participants aim to clarify their understanding and reach a consensus on the correct logical formulation.
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Homework Statement


The rules for a certain frequent-flyer club include the following statements: "Any member who fails to earn any mileage during the first twelve months after enrollment in the program may be removed from the program. Except as otherwise provided, any member who fails at any time to earn mileage for a period of three consecutive years is subject to termination of his or her membership and forfeiture of all accrued mileage. Notwithstanding this provision, no pre-July 1 2014, member who has earned mileage ( other than enrollment bonus) prior to July 1, 2005, shall be subject under this provision to the termination of his or her membership and to the cancellation of mileage accrued prior to July 1 , 2005, until the amount of such mileage falls below 10,000 miles (the amount necessary for the lowest available award under the structure in place as of June 30, 2004), or until December 15, 2015, whichever comes first.

Homework Equations


Let x be a particular member of this club, and let
p= "x fails to earn mileage during the first twelve months after enrollment"
q= "x fails to earn mileage for a period of time of three consecutive years"
r= "x became a member prior to July 1, 2004"
s= "x currently has at least 100,000 miles for pre-July 1,2005,mileage (not including enrollment bonus miles)"
t= "the current date is prior to December 15, 2015"
Use symbols to write the complete condition under which x's membership may be terminated.

The Attempt at a Solution


My answer : [ (p V q) N ~r] V ( s V t) ( N means AND)
What do you think? What's yours?
 
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My solution looks quite different. For a start, it seems to me that condition p is sufficient in itself, so the answer should be of the form pV(...).
 
haruspex said:
My solution looks quite different. For a start, it seems to me that condition p is sufficient in itself, so the answer should be of the form pV(...).
Can you please give your full answer so that I can derive a truth table? Thanks.
 
Ayushi160695 said:
Can you please give your full answer so that I can derive a truth table? Thanks.
This is a homework forum. We do not hand out complete answers.
Do you agree that condition p by itself is a sufficient condition? If so, please make another attempt.
 
haruspex said:
This is a homework forum. We do not hand out complete answers.
Do you agree that condition p by itself is a sufficient condition? If so, please make another attempt.
Okay. I'm sorry. I'm new here.
No, I don't agree since even if p is true but the condition r is true as well then its condition r that matters not p. That is even if the person fails to earn any mileage during the first twelve months, if the person was a member prior to 1 July 2004, then her/his contract will not be terminated as per condition p, provided s or t are false as well.
 
Ayushi160695 said:
No, I don't agree since even if p is true but the condition r is true as well then its condition r that matters not p. That is even if the person fails to earn any mileage during the first twelve months, if the person was a member prior to 1 July 2004, then her/his contract will not be terminated as per condition p, provided s or t are false as well.
Then we read it differently. Here's how I read it. This first statement is absolute and not qualified by anything that comes later:
Ayushi160695 said:
Any member who fails to earn any mileage during the first twelve months after enrollment in the program may be removed from the program.
The next statement gives a second reason for termination, but notes that there are some exceptions to this rule:
Ayushi160695 said:
Except as otherwise provided, any member who fails at any time to earn mileage for a period of three consecutive years is subject to termination o
It then proceeds to describe the exceptions to "this provision":
Ayushi160695 said:
Notwithstanding this provision,
 
haruspex said:
Then we read it differently. Here's how I read it. This first statement is absolute and not qualified by anything that comes later:

The next statement gives a second reason for termination, but notes that there are some exceptions to this rule:

It then proceeds to describe the exceptions to "this provision":
I see. I believe you are right. Then my answer would be : p V [ (q N ~r) V ( s V t) ]
What do you think?
 
Ayushi160695 said:
I see. I believe you are right. Then my answer would be : p V [ (q N ~r) V ( s V t) ]
What do you think?
Ok, we now agree on what happens if p is true. So suppose it is false. For that case you have (q N ~r) V ( s V t). In particular, that says that s and t are each in themselves sufficient basis for termination. That is very odd since they actually contribute to reason for non-termination.
 
haruspex said:
Ok, we now agree on what happens if p is true. So suppose it is false. For that case you have (q N ~r) V ( s V t). In particular, that says that s and t are each in themselves sufficient basis for termination. That is very odd since they actually contribute to reason for non-termination.
Indeed. If both s and t are false, then (s V t) = 0, and if r is false then then ~r is 1. It then depends on q, if q is true, they are terminated, else they are not terminated.
Another case where if s or t is true then ( s V t) = 1, and if r is false then ~r is 1. If q is true or false, they are terminated. But if r is true, then ~r is 0. If q is true or false, then they are terminated again. When q is false, even if r , s or t are true, they should not be terminated. which makes me think that p V [ (q N ~r) V ( s V t) ] is erroneous.
 
  • #10
Ayushi160695 said:
p V [ (q N ~r) V ( s V t) ] is erroneous
Quite so. Any more ideas?
 
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