Write a proof by contraposition for S as a free response. Original Question Prompt in Description.

Question

Let S be the statement: The reciprocal of any irrational number is irrational. (The reciprocal of a nonzero real number x is 1/x.)

Q: Which of the following is the contrapositive of S?

A: If the reciprocal of a real number is not irrational, then the number is not irrational.

Answer 1

Suppose y is irrational. And suppose the reciprocal of y is rational.

Then 1/y = a/b where a, b are integers.

Then y = b/a where a.b are integers.

Then y is rational. But it can't be because it's assumed that it is irrational.

Therefore the reciprocal of y can't be rational so it must be irrational.

Mike