4. For each of the following, first write the argument in symbolic form, and the

Question

4. For each of the following, first write the argument in symbolic form, and then determine if it is valid or invalid. If the argument is valid, prove it using known logical equivalences and inference rules. If it is invalid, demonstrate that by giving a counterexample. (a) If the team could not play good defence, or the forwards did not score enough goals, then the team would not have qualified for the playoffs and fans would have been disappointed. If the the team did not qualify for the playoffs, then tickets would need to be refunde No tickets were refunded. Therefore, the team could play good defence. (b) I cycle to work or ride my motorcycle to work. If I cycle to work, then if I have an early class, I need to hurry. If don't complain, then I have an early class. Therefore, if I don't need to hurry then I don't complain.

Explanation / Answer

(a)

Let p denotes "Team play good defence" and q denotes "Forwards score enough goals"

Let r denotes "Team qualify for playoffs" and s denotes that "Fans were disappointed"

Let t denotes "Tickets were refunded".

We can write all the arguments in symbolic form as,

~p V ~q -> ~r & s

~r -> t

Given, ~t = TRUE

and we need to find the value of p

~t = TRUE  => t = FALSE

~r -> t = TRUE and t = FALSE . So, ~r = FALSE ( p -> q = TRUE and q = FALSE gives p = FALSE)

So, ~r & s = FALSE & s = FALSE (FALSE & a = FALSE)   

or, ~r & s = FALSE

Now, ~p V ~q -> ~r & s = TRUE and ~r & s = FALSE    ( p -> q = TRUE and q = FALSE gives p = FALSE)

so, ~p V ~q = FALSE (a V b = FALSE gives a = b = FALSE)

or, ~p = FALSE and ~q = FALSE

or, p = TRUE

So, Team play good defence

So, the argument is valid.

(b)

Let c denotes "cycle to work" and m denotes "motorcycle to work"

Let e denotes that there is an early class and h denotes "need to hurry"

Let p denotes "need to complain".

We can write all the arguments in symbolic form as,

c -> e

e -> h

~p -> e

Given, ~h = TRUE ad we need to find the value of p

~h = TRUE gives, h = FALSE

e -> h = TRUE and h = FALSE

so, e = FALSE    ( p -> q = TRUE and q = FALSE gives p = FALSE)

~p -> e = TRUE and e = FALSE

so, ~p = FALSE      ( p -> q = TRUE and q = FALSE gives p = FALSE)

so, p = TRUE

Therefore, I need to complain.

So, the argument is invalid.

Related Questions

Navigate

• Browse (All)
• Browse 4
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.