Answers:

(i) Direct Method:

Consider,

q: x is a real number such that x³ + x=0.

r: x is 0.

If q, then r.

Let q be true. Then, x is a real number such that x³ + x = 0

x is a real number such that x(x² + 1) = 0

x = 0

r is true

q is true

q is true and r is true.

Thus, p is true.

(ii) Method of Contrapositive:

Let,

R is not true

x ≠ 0, x∈R

x(x²+1)≠0, x∈R

q is not true

-r = -q

p : q and r is true