(i) Direct Method:
Let us assume that ‘q’ and ‘r’ be the statements given by
q: x is a real number such that x3 + 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 x3 + x = 0
x is a real number such that x(x2 + 1) = 0
x = 0
r is true
Thus, q is true
Therefore, q is true and r is true.
Hence, p is true.
(ii) Method of Contrapositive:
Let r be false. Then,
R is not true
x ≠ 0, x∈R
x(x2+1)≠0, x∈R
q is not true
Thus, -r = -q
Hence, p : q and r is true
(iii) Method of Contradiction:
If possible, let p be false. Then,
P is not true
-p is true
-p (p => r) is true
q and –r is true
x is a real number such that x3+x = 0and x≠ 0
x =0 and x≠0
This is a contradiction.
Hence, p is true.
Answered by Sakshi | 1 year agoDetermine whether the argument used to check the validity of the following statement is correct: p: “If x2 is irrational, then x is rational.” The statement is true because the number x2 = π2 is irrational, therefore x = π is irrational.
Which of the following statements are true and which are false? In each case give a valid reason for saying so
(i) p: Each radius of a circle is a chord of the circle.
(ii) q: The centre of a circle bisect each chord of the circle.
(iii) r: Circle is a particular case of an ellipse.
(iv) s: If x and y are integers such that x > y, then – x < – y.
(v) t: \( \sqrt{11}\) is a rational number.
By giving a counter example, show that the following statement is not true. p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.”
Show that the following statement is true “The integer n is even if and only if n2 is even”
Show that the following statement is true by the method of the contrapositive p: “If x is an integer and x2 is odd, then x is also odd.”