Discrete Mathematics Thegatebook

Question

Q1. Which of the following arguments are invalid?
S1: If it rains Erick will be sick
It did not rain
Erick was not sick
S2: If I study then I will not fail mathematics
If I do not play basketball, then I will study
But I failed mathematics
Therefore I must have played basketball
a) Only S1 b) Only S2 c) both S1 and S2 d) neither S1 nor S2

Answer 1

It is always preferable to replace the english clauses with some predicates ..Let us see through examples ..For first one I m assuming as :

p : It rains 

q : Erick will be sick.

So the statement "  If it rains Erick will be sick " is translated as p --> q 

Now it is given it did not rain meaning we have : ∼p 

And the conclusion is given to be "Erick is not  sick." which is ∼q 

So with the given set of statements we can say that p --> q holds but it does not mean that  ∼p --> ∼q  should hold necessarily..Hence the conclusion "Erick is not  sick." which is ∼q  is invalid..

Hence S1 is invalid.

Now for S2 , we have : 

p : I study 

q : I will not fail mathematics

r : I do not play basketball

So ,

"If I study then I will not fail mathematics" is translated as : p --> q

"If I do not play basketball, then I will study" is translated as : r --> p

N it is given he failed in maths meaning  : ∼q  is given

So from contrapositive statement of 1st one we get ∼q --> ∼p

    from contrapositive statement of 2nd one we get ∼p --> ∼r

So ∼r suggests that he must have played basketball..

Hence S2 is valid..

Hence B) is the correct answer..

Comments

Thank you brother