The given compound statement is of the from “if p then q”.
P: x ∈ R such that x3 + 4x = 0
q: x = 0
1. Direct method:
We assume that p is true then
x ∈ R such that x3 + 4x = 0
x ∈ R such that x(x2 + 4) = 0
x ∈ R such that x = 0 or x2 + 4 = 0
⇒ x = 0
⇒ q is true.
So when p is true, q is true.
Thus the given compound statement is true.
2. Method of contradiction:
We assume that p is true and q is false, then x ≠ 0
x ∈ R such that x3 + 4x = 0
x ∈ R such that x(x2 + 4) = 0
x ∈ R such that x = 0 or x2 + 4 = 0
⇒ x = 0
Which is a contradiction. So our assumption that x ≠ 0 is false. Thus the given compound statement is true.
3. Method of contrapositive.
We assume that q is false, then x ≠ 0
⇒ x ∈ R such that x3 + 4x ≠ 0
⇒ q is false
So when q is false, p is false.
Thus the given compound statement is true