Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.)

1. Suppose nZ. If n² is odd, then n is odd.

2. Suppose x R. If x³ - x>0 then x>-1.

3.Suppose x, y, z Z and x0. If xyz, then xy and x z. ( this symobl indicates not divisible)

Prove the following statements using either direct or contrapositive proof.

4. If a Z and a 1 (mod 5), then a²1 (mod 5).

5. Let a Z, n N. If a has remainder r when divided by n, then a r (mod n).

6. If ab (mod n) and cd (mod n), then ac bd (mod n).

7. If a b (mod n), then gcd(a,n) = gcd(b,n).

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.)

8. Prove that 6 is irrational.

9. There exist no integers a and b for which 21a+30b=1.

10.If A and B are sets, then A(B-A) = .

Prove the following statements using any method like contrapositive proof, proof by contradiction, direct or contrapositive proof.

11. We say that a point P = (x, y) in R² is rational if both x and y are rational. More precisely, P is rational if P = (x, y) Q². An equation F(x, y) = 0 is said to have a rational point if there exists X₀, Y₀ Q such that F(X₀, Y₀) = 0. For example, the curve X² +Y² - 1 = 0 has rational point (X₀, Y₀) = (1,0). Show that the curve X² +Y² - 3 = 0 has no rational points.