logical connective problem?

Question

What do you think of this statement?

"If it is sunny or if I have no work to do, then I am happy. I am happy and it is sunny. Therefore I have no work to do."

Let me define each of these statements:
Let p: "It is sunny"
Let q: "I have work to do"
Let r: "I am happy"

My answer is:
{ [ (p V ~q) => r ] Λ (r Λ p) } => ~q
Is my answer right? My teacher told me that it is right, but I have doubts in it. Can this statement "IF it is sunny or IF I have no work to do, then I am happy." be said as: (p => r) Λ (~q => r)? I tried comparing their results using a TRUTH TABLE but the outputs are not the same.

For those who are willing to help and did not encounter LOGICAL CONNECTIVES: The symbol ' V ' means OR. The symbol ' Λ ' means AND. The symbol ' => ' means imply (conditional purposes only)

Answer 1

"If it is sunny or if I have no work to do, then I am happy. I am happy and it is sunny. Therefore I have no work to do."

(p v ~q) => r

Given that you already know it is sunny, you know you will be happy from the above REGARDLESS of whether or not you have work to do because (p v ~q) will be true. You cannot conclude anything about whether or not you have work to do because in either case, p v ~q will be true, thus making r true.

Answer 2

i) "If it is sunny or if I have no work to do, then I am happy. I am happy and it is sunny. Therefore I have no work to do."

i) { [ (p V ~q) => r ] Λ (r Λ p) } => ~q

/******************************************/


ii) "If it is sunny AND if I have no work to do, then I am happy. I am happy and it is sunny. Therefore I have no work to do."

ii) { [ (p Λ ~q) => r ] Λ (r Λ p) } => ~q
/******************************************/

In my estimation
i) is FALSE
ii) is TRUE

Answer 3

your answer is right but the second part isn't the second part would be
pV~q=>r