Solution for Section 3.6 Question 3 (by contraposition)

3.. We are asked to prove, using the method of contraposition, the statement: "rQ with r non-zero, if (s is irrational), then (r·s is irrational).

The contrapositive of the above statement is:  "rQ with r non-zero, if (r·s is rational), then (s is rational).

Now prove the contrapositive by a direct proof.

Proof Suppose that r is a non-zero rational number and the product r·s is also rational.  Thus,  there exist integers a, b, c and d such that  r = a / b and  r·s = c / d,
where a, b, d 0. 

Since a, b, c and d are all integers,  c·b  and a·d   are also integers. Also, since a, b and d are non-zero,   a·d   is non-zero. Hence s can be written as a quotient (fraction) of integers. Therefore s is rational.

Since the contrapositive statement is equivalent to the original statement, we can now conclude that "rQ with r non-zero, if (s is irrational), then (r·s is irrational).

Back to Section 3.6