15. Show that the statement “For any real numbers a and b, a2 = b2 implies that a = b” is not true by giving a counter-example.
Answer :
The given statement can be written in the form of “if-then” as follows.
If a and b are real numbers such that a2 = b2 , then a = b.
Let p: a and b are real numbers such that a2 = b2 .
q: a = b
The given statement has to be proved false. For this purpose, it has to be proved that if p, then ∼q.
To show this, two real numbers, a and b, with a2 = b2 are required such that a ≠ b. Let a = 1 and b = –1 a2 = (1)2 = 1 and b2 = (– 1)2 = 1
∴ a2 = b2
However, a ≠ b
Thus, it can be concluded that the given statement is false.
16. Show that the following statement is true by the method of contrapositive. p: If x is an integer and x2 is even, then x is also even.
Answer :
p: If x is an integer and x2 is even, then x is also even.
Let q: x is an integer and x2 is even. r: x is even.
To prove that p is true by contrapositive method, we assume that r is false, and prove that q is also false.
Let x is not even.
To prove that q is false, it has to be proved that x is not an integer or x2 is not even.
x is not even implies that x2 is also not even.
Therefore, statement q is false.
Thus, the given statement p is true.
17. By giving a counter example, show that the following statements are not true.
(i) p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.
(ii) q: The equation x2 – 1 = 0 does not have a root lying between 0 and 2.
Answer :
(i) The given statement is of the form “if q then r”.
q: All the angles of a triangle are equal. r: The triangle is an obtuse-angled triangle.
The given statement p has to be proved false.
For this purpose, it has to be proved that if q, then ∼r.
To show this, angles of a triangle are required such that none of them is an obtuse angle.
It is known that the sum of all angles of a triangle is 180°. Therefore, if all the three angles are equal, then each of them is of measure 60°, which is not an obtuse angle. In an equilateral triangle, the measure of all angles is equal. However, the triangle is not an obtuse-angled triangle.
Thus, it can be concluded that the given statement p is false.
(ii) The given statement is as follows. q:
The equation x2 – 1 = 0 does not have a root lying between 0 and 2.
This statement has to be proved false. To show this, a counter example is required.
Consider x2 – 1 = 0 x2 = 1 x = ± 1
One root of the equation x2 – 1 = 0, i.e. the root x = 1, lies between 0 and 2.
Thus, the given statement is false.
18. 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 bisects 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: √11is a rational number.
Answer :
(i) The given statement p is false. According to the definition of chord, it should intersect the circle at two distinct points.
(ii) The given statement q is false. If the chord is not the diameter of the circle, then the centre will not bisect that chord. In other words, the centre of a circle only bisects the diameter, which is the chord of the circle.
(iii) The equation of an ellipse is, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
If we put a = b = 1, then we obtain x2 + y2 = 1, which is an equation of a circle
Therefore, circle is a particular case of an ellipse.
Thus, statement r is true.
(iv) x > y
⇒ –x < –y (By a rule of inequality)
Thus, the given statement s is true.
(v) 11 is a prime number and we know that the square root of any prime number is an irrational number. Therefore, √11 is an irrational number.
Thus, the given statement t is false.
19. Write the negation of the following statements:
(i) p: For every positive real number x, the number x – 1 is also positive.
(ii) q: All cats scratch.
(iii) r: For every real number x, either x > 1 or x < 1.
(iv) s: There exists a number x such that 0 < x < 1.
Answer :
(i) The negation of statement p is as follows.
There exists a positive real number x, such that x – 1 is not positive.
(ii) The negation of statement q is as follows.
There exists a cat that does not scratch.
(iii) The negation of statement r is as follows. There exists a real number x, such that neither x > 1 nor x < 1.
(iv) The negation of statement s is as follows. There does not exist a number x, such that 0 < x < 1.
20. State the converse and contrapositive of each of the following statements:
(i) p: A positive integer is prime only if it has no divisors other than 1 and itself.
(ii)q: I go to a beach whenever it is a sunny day.
(iii) r: If it is hot outside, then you feel thirsty.
Answer :
(i) Statement p can be written as follows.
If a positive integer is prime, then it has no divisors other than 1 and itself.
The converse of the statement is as follows.
If a positive integer has no divisors other than 1 and itself, then it is prime.
The contrapositive of the statement is as follows. If positive integer has divisors other than 1 and itself, then it is not prime.
(ii) The given statement can be written as follows.
If it is a sunny day, then I go to a beach.
The converse of the statement is as follows.
If I go to a beach, then it is a sunny day. The contrapositive of the statement is as follows. If I do not go to a beach, then it is not a sunny day.
(iii) The converse of statement r is as follows.
If you feel thirsty, then it is hot outside.
The contrapositive of statement r is as follows.
If you do not feel thirsty, then it is not hot outside.
21. Write each of the statements in the form “if p, then q”.
(i) p: It is necessary to have a password to log on to the server.
(ii) q: There is traffic jam whenever it rains.
(iii) r: You can access the website only if you pay a subscription fee.
Answer :
(i) Statement p can be written as follows. If you log on to the server, then you have a password.
(ii) Statement q can be written as follows. If it rains, then there is a traffic jam.
(iii) Statement r can be written as follows. If you can access the website, then you pay a subscription fee.