Mathematical Reasoning Exercise 05
Question:
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:

Let p be the statement “If x is an integer and x2 is even then x is also even”.

Rewrite the statement in the form of a contrapositive: “If x is an integer and x is not even then x2 is not even”.

To show that this statement is true, assume that x is an integer and x is not even.

Since x is not even, x must be odd.

Since x is odd, x2 must be odd as well.

Therefore, if x is an integer and x is not even then x2 is not even.

This shows that the statement is true by the method of contrapositive.
Question:
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 : √11 is a rational number
Answer:
(i) p : True. Each radius of a circle is a line segment joining the centre of the circle to any point on the circumference of the circle. Since a chord is also a line segment joining any two points on the circumference of the circle, each radius of a circle is also a chord of the circle.
(ii) q : False. The centre of a circle does not bisect each chord of the circle. It only bisects the chord that passes through the centre of the circle.
(iii) r : False. A circle is not a particular case of an ellipse. A circle is a special case of an ellipse where the two foci are at the same point.
(iv) s : True. If x and y are integers such that x>y then −x<−y. This is because when two numbers are multiplied by a negative number, the result is the opposite of the order of the original numbers.
(v) t : False. √11 is an irrational number, not a rational number. This is because the decimal expansion of √11 is nonterminating and nonrepeating.
Question:
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 x^2−1 =0 does not have a root lying between 0 and 2.
Answer:
(i) Counter example: Consider a triangle ABC with all angles equal to 60°. This triangle is an acute angled triangle and not an obtuse angled triangle. Therefore, the statement p is not true.
(ii) Counter example: Consider the equation x^2−1 = 0. This equation has a root at x = 1, which lies between 0 and 2. Therefore, the statement q is not true.
Question:
Show that the statement “For any real numbers a and b, a^2=b^2 implies that a=b” is not true by giving a counterexample
Answer:
Counterexample: Let a = 2 and b = 2.
Then a^2 = (2)^2 = 4 = b^2 = 2^2 = 4
However, a ≠ b, since a = 2 and b = 2.
Therefore, the statement “For any real numbers a and b, a^2=b^2 implies that a=b” is not true.
Question:
Show that the statement p : “If x is a real number such that x^3+4x=0 then x is 0” is true by (i) direct method (ii) method of contradiction (iii) method of contrapositive
Answer:
(i) Direct Method:
Let x be any real number such that x^3+4x=0.
Therefore, x^3=4x
Dividing both sides by x, we get
x^2=4
Taking the square root of both sides, we get
x=2
Substituting x=2 in the given equation, we get
(2)^3+4(2)=0
Therefore, the statement p is true.
(ii) Method of Contradiction:
Assume that the statement p is false. That is, assume that there exists a real number x such that x^3+4x=0 but x is not 0.
Therefore, x^3=4x
Dividing both sides by x, we get
x^2=4
Taking the square root of both sides, we get
x=2
Substituting x=2 in the given equation, we get
(2)^3+4(2)=0
This contradicts our assumption that x is not 0. Therefore, the statement p is true.
(iii) Method of Contrapositive:
The contrapositive of the statement p is:
p’: If x is not 0, then x^3+4x is not equal to 0.
Let x be any real number such that x is not 0.
Therefore, x^3+4x is not equal to 0.
This proves that the statement p’ is true. Therefore, the statement p is true.