Question

Show that the statement
: "If is a real number such that  then is " is true by 
(i) direct method
(ii) method of contradiction
(iii) method of contrapositive

Medium

Solution

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: "If is a real number such that  then is "
Let is a real number such that 
is
(i) To show that statement is true we assume that is true and then show that is true.
Therefore let statement be true.



However since is real it is .
Thus statement is true
Therefore the given statement is true.
(ii) To show statement to be true by contradiction we assume that is not true.
Let be a real number such that  and let is not .
Therefore 

or 
or 
However is real. Therefore which is a contradiction since we have assumed that is not .
Thus the given statement p is true.
(iii) To prove statement to be true by contrapositive method we assume that is false and prove that must be false.
Here is false implies that it is required to consider the negation of statement . This obtains the following statement,

It can be seen that  will always be positive.
x implies that the product of any positive real number with is not zero.
Let us consider the product of with 


This shows that statement is not true.
Thus it has been proved that 

Therefore the given statement is true.

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Solution to Question ID 418283
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