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Question -

Show that the statement
p : тАЬIf x is a real number such that x3 + x = 0, then x is 0тАЭ is true by
(i) Direct method
(ii) method of Contrapositive
(iii) method of contradiction



Answer -

(i) DirectMethod:

Let us assume that тАШqтАЩ and тАШrтАЩ be the statements given by

q: x is a real number such that x+ x=0.

r: x is 0.

The given statement can be written as:

if q, then r.

Let q be true. Then, x is a real number such that x+x = 0

x is a real number such that x(x+ 1) = 0

x = 0

r is true

Thus, q is true

Therefore, q is true and r is true.

Hence, p is true.

(ii) Methodof Contrapositive:

Let r be false. Then,

R is not true

x тЙа 0, xтИИR

x(x2+1)тЙа0, xтИИR

q is not true

Thus, -r = -q

Hence, p : q and r is true

(iii) Methodof Contradiction:

If possible, let p be false. Then,

P is not true

-p is true

-p (p => r) is true

q and тАУr is true

x is a real number such that x3+x = 0and xтЙа 0

x =0 and xтЙа0

This is a contradiction.

Hence, p is true.

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