Question 1:Show that the statementp: “If x is a real number such that x3 4x = 0, then x is 0” is true by(i) direct method(ii) method of con…

Answer:
p:
“If x is a real number such that x3 4x = 0, then x is 0”.
Let q: x is a real number such that x3 4x = 0
r: x is 0.
(i) To show that
statement p is true, we assume that q is true and then show that r is
true.
Therefore,
let statement q be true.
∴ x3 4x = 0
x (x2 4) = 0
⇒ x = 0 or x2
4 = 0
However,
since x is
real, it is 0.
Thus,
statement r is true.
Therefore,
the given statement is true.
(ii) To show
statement p to be true by contradiction, we assume that p is not true.
Let x be a real number such that x3 4x = 0 and let x is not 0.
Therefore, x3 4x = 0
x (x2 4) = 0
x = 0 or x2 4 = 0
x = 0 or x2 = – 4
However, x is real. Therefore, x =
0, which is a contradiction since we have assumed that x is not 0.
Thus,
the given statement p is true.
(iii) To prove
statement p to be true by contrapositive method, we assume that r is false and prove that q must be false.
Here, r is false implies that it is required to consider the negation of
statement r.
This obtains the following statement.
∼r: x is not 0.
It
can be seen that (x2 4) will always be positive.
x ≠ 0 implies that the product of any positive real number with x is not zero.
Let
us consider the product of x with (x2 4).
∴ x (x2 4) ≠ 0
⇒ x3 4x ≠ 0
This
shows that statement q is not true.
Thus,
it has been proved that
∼r ⇒
∼q
Therefore,
the given statement p is true.