Show that the statement $p:$ "If $x$ is a real number such that $x^{3}+4 x=0$, then $x$ is 0 " is true by
(i) direct method,
(ii) method of contradiction,
(iii) method of contrapositive

Answer

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Geometry

NCERT Class 11 - Math

Chapter 14

Mathematical Reasoning

Section 5

Implications

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Video Transcript

show that the statement P f x is a real numbers that died Xq plus four X is equal to zero. Then X zero is true. By using the method by using direct method method of contradiction and method of contra positive. So here we have given the statement P. That is if X is a real numbers such that X Q plus four X is equal to zero, then X is zero. Now let the statement Q. That is X is a real number. What such bad X cubed plus four, X is equal to zero. And let's take the statement are that is X zero. In part one. We have to show that the statement B is true. We assume that Q is true and then show that are is true. Dead statement QB true. So we have x cubed plus four AIDS. It's equals to zero. Now by taking X com and we get X into x squared plus four is equal to zero. Now putting each factor is equal to zero. We get X equals to zero. All X squared plus four is equal to zero. However, since X is real it is zero. The statement are is true. Their fall, the given statement is true. Now in part two we have to show that he to be true. By contradiction. We assume that the statement B is not true here. Let expiry real numbers at that X cubed plus four, X is equal to zero And let access not equals to zero. Therefore we get X cubed plus four. X is equal to zero. Taking X comment. We get X men do X squared plus four is equals to zero. Now putting each factories equals to zero. We get X. S equals to zero. All X squared plus four is equals to zero. How we get X. S equals to zero. All here we have X squared equals two minus four. However X is real, therefore XS equals to zero. Which is contradiction. Since we have assumed that X is not equal to zero, there's the given statement B is true. Sure. In part three to prove statement pay to be truly by contra positive method, we assume that our s false and prove that you must be false here are exposed implies that it is required to consider the negation of statement. Are this obtains the following statement? Our is not zero. It can be seen that X squared plus full will always be positive access not equals to zero, implies that the product of any positive real number with X is not zero. Let us consider the product of eggs with X squared plus four, It's not equals to zero. So we have xQ plus four, eggs is not equals to zero. This shows that statement Q is not true. Does it has been proved that negation of our employees negation of cube. Hence the given statement B is true.

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