Question

1. Let x be an irrational real number. (a) Explain why 22 is not guaranteed to be irrational. (b) Prove that 22 is irrational or 23 is irrational.

Answer 1

holds. a) Suppose, & irrational implies ² irrational Them, contrapositive statement is! 1 x2 rational implies a is rational Non, (2) 2 22 is mentional. Then, above implies a Na is rational. But, we know, sz is innational. . Contradiction. I Then, a innational does not grantee x² irrational. b) Let, a is irrational. Need to show, e² is innational or x3 is irrational. • Suppose, az is rational. O Then, a irrational =) s. x2=X² is innational. C Cas product of non- zero rational with menzero irrational is irrational , . . Suppose , x3 is rational. Then, az rational implics, se ex is national Contradiction .. x² is in rational - O,☺ => n is irrational then x² or a3 is irrationala Please uprote.