Answer
the supposition is false and the statement is true that The product of any nonzero rational number and any irrational
number is irrational.
Work Step by Step
The product of any nonzero rational number and any irrational number is irrational.
$Proof$: Suppose not. That is, suppose ∃ a nonzero rational number x and an irrational number y such that xy is rational.
-[We must derive a contradiction.]
- By definition of rational, x = a/b and xy = c/d
for some integers a, b, c, and d with b $\ne$ 0 and d $\ne$ 0. Also a $\ne$ 0 because x is nonzero.
- By substitution, xy = (a/b)y = c/d. Solving for y gives y = bc/ad. Now bc and ad are integers (being products of integers)
and ad $\ne$ 0 (by the zero product property).
Thus,
- by definition of rational, y is rational, which contradicts the supposition that y is irrational.
[Hence the supposition is false and the statement is true.]
