Answer to Question #273647 in Discrete Mathematics for waliul

Question

Use rules of inference to show that the hypotheses “If the weather is not

too hot or not too cold, then the game will be held and a prize-giving

ceremony will occur,” “If the game is held then the VC will give a speech,”

“The VC did not give a speech,” imply the conclusion “The weather was

too hot.”

Answer 1

Let,

p = Weather is too hot

q = Weather is too cold

r = game will held

s = VC will give a speech

As per the statements,

"If the weather is not too hot or not too cold" = ~p v ~q

"then the game will held and a prize giving ceremony will occur" = (~p v ~q) -> r

"If the game is held then the VC will give a speech" = r -> s

Finally,

"The VC didn't give a speech" = ~s

Conclusion is "The weather was too hot" = ~s => p

Writing the above statements altogether as,

~s => p

s is being tend from r, so back substituting s to r as,

~r => p

r is being tend from ~p v ~q, so back substituting r to ~p v ~q as,

~(~p v ~q) => p

Therefore, p ^ q => p

Let us construct Truth Table for the above statement p ^ q => p

(p ^ q) => p is one of the rules under Rules of Inference, the rule name is Simplification.

p q p ^ q (p ^ q) -> p

T T T T

T F F T

F T F T

F F F T

The values of the last column says all the conditions are True, therefore the statement (p ^ q) => p holds True.