Exercise

Use a direct proof to show that the sum of two odd integers is even.

Show that the square of an even number is an even number using a direct proof.

Prove that the sum of two rational numbers is a rational number using a direct proof.

Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational.

Prove or disprove that the product of two irrational numbers is irrational.

Prove that if x is irrational then 1/x is irrational. Proof by contraposition: If 1/x were rational, then by definition

Use a proof by contraposition to show that If x+y≥2, where x and y are real numbers , then x≥1 or y≥1.