Punjab State Board PSEB 11th Class Maths Book Solutions Chapter 14 Mathematical Reasoning Ex 14.3 Textbook Exercise Questions and Answers.
PSEB Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning Ex 14.3
Question 1.
For each of the following compound statements first identify the connecting words and then break it into component statements.
(i) All rational numbers are real and all real numbers are not complex.
(ii) Square of an integer is positive or negative.
(iii) The sand heats up quickly in the Sun and does not cool down fast at night.
(iv) x = 2 and x = 3 are the roots of the equation 3x2 – x – 10 = 0.
Answer.
(i) Here, the connecting word is ‘and’.
The component statements are as follows.
p : All rational numbers are real.
q : All real numbers are not complex.
(ii) Here, the connecting word is ‘or’.
The component statements are as follows,
p : Square of an integer is positive.
q : Square of an integer is negative.
(iii) Here, the connecting word is ‘and’.
The component statements are as follows.
p : The sand heats up quickly in the sun.
q : The sand does not cool down fast at night.
(iv) Here, the connecting word is ‘and’.
The component statements are as follows.
p : x = 2 is a root of the equation 3x2 – x -10 = 0
q : x = 3 is a root of the equation 3x2 – x -10 = 0
Question 2.
Identify the quantifier in the following statements and write the negation of the statements.
(i) There exists a number which is equal to its square.
(ii) For every real numbers, x is less than x + 1.
(iii) There exists a capital for every state in India.
Answer.
(i) Quantifier : There exists.
p : There exists a number which is equal to its square
not p : There does not exist a number which is equal to its square.
(ii) Quantifier : For every
p : For every real number x, x is less than x + 1
~p : For every real number x, x is not less than x + 1
(iii) Quantifier : There exists
p : There exists a capital for every state of India.
~ p : There does not exist a capital for every state of India.
Question 3.
Check whether the following pair of statements are negation of each other. Give reasons for the answer.
(i) x + y = y + x is true for every real numbers x andy.
(ii) There exists real number x and y for which x + y = y + x.
Answer.
Let p: x + y = y + x is true for every real numbers x and y.
q : There exists real numbers x and y for which x + y = y + x.
Now ~ p : There exists real numbers x and y for which x + y ≠ y + x. Thus ~ p ≠ q.
Question 4.
State whether the “Or” used in the following statements is exclusive “or” inclusive. Give reasons for your answer.
(i) Sun rises or Moon sets.
(ii) To apply for a driving licence, you should have a ration card or a passport.
(iii) All integers are positive or negative.
Answer.
(i) Here, “or” is exclusive because it is not possible for the Sun to rise and the Moon to set together.
(ii) Here, “or” is inclusive since a person can have both a ration card and a passport to apply for a driving licence.
(iii) Here, “or” is exclusive because all integers cannot be both positive and negative.