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Question

Consider the statement :

P:if x a real number such that x3+4x=0, then x=0

Prove that p is a true statement , using: (i) direct method (ii) method of contradiction (iii) method of contrapositive

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Solution

(i) Direct method:

let x3+4x=0, where xR,then,

x3+4x=0x(x2+4)=0

x=0[x2+40 for xR]

Hence, p is a true statement.

(ii) Method of contradiction

if possible,let(x3+4x=0 and x0 then,

x2(x2+4)=0 and x0x2+4=0

But, this is a contradiction, sincex2+40 for xR

Since, the contradiction arises by assuming that

x2+4=0 and x0,so x=0

Hence,x3+4x=00 is a true statement.

(iii) Method contrapositive:

We have to porve that x3+4x=0x=0

Let P:x3+4x=0 and q:x=0.


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