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Mathematical Reasoning (Ex – 14.2)
Question 1.
Write the negation of the following statements:
(i) Chennai is the capital of Tamil Nadu.
(ii) √2 is not a complex number.
(iii) All triangles are not equilateral triangle.
(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: √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.
