1.(ii)    Show that the statement
            p: “If x is a real number such that x^3 + 4x = 0, then x is 0” is true by

            (ii) method of contradiction

Answers (1)

If x is a real number such that x^3 + 4x = 0, then x is 0 : (if p then q)

p: x is a real number such that x^3 + 4x = 0.

q: x is 0.

In order to prove the statement “if p then q” 

Contradiction:  By assuming that p is true and q is false.

So,

p is true:  There exists a real number x such that x^3 + 4x = 0

q is false: x \neq 0

Now, x^3 + 4x = 0 \implies x(x^2 + 4) = 0

\implies x = 0\ or\ (x^2 + 4)= 0

\implies x = 0\ or\ x^2 = -4\ (not\ possible)

Hence, x = 0

But we assumed x \neq 0. This contradicts our assumption.

Therefore q is true.

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