Question

.Write each of the following arguments in symbolic form. Then establish the validity of the argument or give a counter-example to show that it is invalid

a) If Dominic goes to the racetrack, then Helen will be mad. if Ralph plays cards all night, then Carmela will be mad. If either Helen or Carmela gets mad, then Veronica (their attorney) will be notified. Veronica has not heard from either of these two clients. Consequently, Dominic didn’t make it to the racetrack and Ralph didn’t play cards all night.

b) If there is a chance of rain or her red headband is missing, then Lois will not mow her lawn. Whenever the temperature is over 80 F, there is no chance for rain. Today the temperature is 85 F and Lois is wearing her red headband. Therefore (sometime today) Lois will mow her lawn.

Answer 1

(a)

Let us consider

p : Dominic goes to the racetrack

q : Helen goes mad

r : Ralph is playing cards all night

s : Carmela will go mad

t :  Veronica will be notified.

If Dominic goes to the racetrack, then Helen will be mad.

Ralph plays cards all night, then Carmela will be mad.

Either Carmela or Helen gets mad then their lawyer Veronica will be notified

Veronica has not heard from either of these two clients

Dominic did not make it to the racetrack and Ralph didn’t play cards all night

Therefore,

. 

Answer 2

(b)

Let us consider

p: There is a chance of rain.

q: Lois’ red headband is missing.

r: Lois will mow her lawn.

s: The temperature is over 20ºC.

There is a chance of rain or her hairband is missing.

There is a chance of rain or her hairband is missing, then Lios wil mow her lawn.

Whenever the temperature is above 20°C then it wouldn't be raining.

Today the temperature is 22°C and Lios is wearing her hairband.

Lios will mow her lawn.

Therefore

.