I saw this question where I had to prove/disprove that:
Ques. Product of two irrational number is irrational.
I tried 'Proof by Contraposition'.
Product of two irrational number is irrational.
p : Product of two irrational number
q : Irrational number.
Thus, given statement is : p -> q
Contraposition of p : ¬q -> ¬p
Rational number -> Can be broken down into product of two rational number.
Proof :
Let m be a rational number such that m = p/q.
Then I can always write m as (p/1)*(1/q)
where (p/1) and (1/q) are both rational numbers. Hence proved.
But it turns out that books disproves the statement saying $\sqrt2\cdot\sqrt2=2$ which is a rational number and hence Product of two irrational number need not always be irrational. Which I find convincing.
Can someone please point out where am I going wrong in my proof?
Best Answer
The negation of the assertion [Is the product of two irrational numbers] is the assertion [Is not the product of two irrational numbers]. There is no a priori reason to expect that the assertion [Is not the product of two irrational numbers] is equivalent to the assertion [Is the product of two rational numbers] (and in fact these last two are not equivalent).