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in Mathematical Reasoning by (15.2k points)

Show that the statement p: “If x is a real number such that x3 + 4x = 0 then x is 0” is true by

1. Direct method. 

2. Method of contradiction. 

3.Method of contrapositive.

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1 Answer

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by (15.7k points)

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

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