Indirectly prove this:?

To prove indirectly, rewrite your conditional statement as follows: p --> q becomes p and ~q; we show the p and ~q is false which means that p --> q must be true.

So, p and ~q is:

The product of 2 natural numbers is greater than 100, and both of the numbers are less than or equal to 10. We show that this statement results in a contradiction.

Let x,y be two natural numbers such that xy > 100 and suppose that both x and y are less than or equal to 10. We consider three cases.

Case 1. x = y = 10. Then, xy = (10)(10) = 100. But this contradicts out assumption that xy > 100.

Case 2. Both x < 10 and y < 10. The xy < 100. Again, we have a contradiction.

Case 3. Exactly one of x or y is less than 10.

Subcase 1. x < 10 and y = 10. Here, xy < 100. We have our necessary contradiction.

Subcase 2 is similar to subcase 1 and is therefore omitted.

Since all three cases for x and y result in a contradiction, it is not true that both x and y are less than or equal to 10; that is, at least one of x or y must be greater than 10.