Answer to Question #178304 in Discrete Mathematics for Faizan

Question

Consider

Premises: If there was a cricket match, then traveling was difficult.

If they arrived on time, then traveling was not difficult.

They arrived on time.

Conclusion: There was no cricket match.

Determine whether the conclusion follows logically from the premises. Explain by

representing the statements symbolically and using rules of inference

Answer 1

Let us use the following notation:

$p=$ "There was a cricket match", $q=$ "Traveling was difficult", $r=$ "They arrived on time".

Then the premises are $p\to q,\ r\to\overline{q}$ and $r$. The conclusion is $\overline{p}.$

Using to the premises $r\to \overline{q}$ and $r$ the rule of inference Modus Ponens, we conclude $\overline{q}$. Then using to $p\to\ q$ and $\overline{q}$ the rule of inference Modus Tollens, we have the conclusion $\overline{p}$. Therefore, the conclusion follows logically from the premises.