Chapter 14 Mathematical Reasoning Ex – 14.2

Question 1.

Write the negation of the following statements:

(i) Chennai is the capital of Tamil Nadu.

(ii) \sqrt { 2 }  is not a complex number.

(iii) All triangles are not equilateral triangles.

(iv) The number 2 is greater than 7.

(v) Every natural number is an integer.

Solution:

(i) Negation of statement is: Chennai is not the capital of Tamil Nadu.

(ii) Negation of statement is: \sqrt { 2 }  is a complex number.

(iii) Negation of statement is: All triangles are equilateral triangles.

(iv) Negation of statement is: The number 2 is not greater than 7.

(v) Negation of statement is: Every natural number is not an integer.

Question 2.

Are the following pairs of statements negations of each other:

(i) The number x is not a rational number.
The number x is not an irrational number.

(ii) The number x is a rational number.
The number x is an irrational number.

Solution:

(i) Let p: The number x is not a rational number.
q: The number x is not an irrational number.
Now, ~p: The number x is a rational number.

~q: The number x is an irrational number.
∴ ~p = q and ~q = p
Thus, p and q are negations of each other.

(ii) Let p: The number x is a rational number.
q: The number x is an irrational number.
Now, ~p: The number x is not a rational number.

~q: The number x is not an irrational number.
∴ ~p = q and ~q = p
Thus, p and q are negations of each other.

Question 3.

Find the component statements of the following compound statements and check whether they are true or false.

(i) Number 3 is prime or it is odd.

(ii) All integers are positive or negative.

(iii) 100 is divisible by 3,11 and 5.

Solution:

(i) The component statements are:
p: Number 3 is prime
q: Number 3 is odd.
Both the component statements p and q are true.

(ii) The component statements are:
p: All integers are positive.
q: All integers are negative.
Both the component statements p and q are false.

(iii) The component statements are:
p: 100 is divisible by 3.
q: 100 is divisible by 11.
r: 100 is divisible by 5.
The component statements p and q are false whereas r is true.