Kenneth Rosen Edition 7 Exercise 1.6 Question 6 (Page No. 78)

Question

Use rules of inference to show that the hypothesis

" If it doesn't rain or it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on ".

"If the sailing race is held, then the trophy will be awarded"

"The trophy was not awarded".

Imply the conclusion "It rained".

Answer 1

Using rules of inference

Let r be the proposition “It rains,”

let f be the proposition “It is foggy,”

let s be the proposition “The sailing race will be held,”

let l be the proposition “The life saving demonstration will go on,” and

let t be the proposition “The trophy will be awarded.”

 premises (¬r ∨¬f) → (s ∧ l),

s → t, and

¬t.

We want to conclude r.

1. ¬t

2. s →t

3. ¬s ( modus tollen 1&2)

4. (¬r ∨¬f) → (s∧l)

 5. (¬(s ∧l)) →¬(¬r ∨¬f)  ( contrapositive)

6. (¬s∨¬l) → (r∧f)

7. ¬s∨¬l (addition from 3)

8. r ∧f (modus ponens)

9. r(conclusion)

Answer 2

p - it is rain.

q - it is foggy.

r - sailing race held.

s - life demo will go on.

t - trophy awarded.

----------------

(~p Λ ~q) -> (r ^ s)

r -> t

~t

--------------------

∴ p (our conclusion)

now using rules of inference, we will get(refer 1.5 section DM Krosen book)

(((¬pV¬q)→(rΛs))Λ(r→t)Λ(¬t))→p it should be tautology.

It is now preferred way but we can made truth table and can check that it is tautology.

To make trutu table for above boolean expression.

turner.faculty.swau.edu/mathematics/materialslibrary/truth/