GSEB Solutions Class 11 Maths Chapter 14 Mathematical Reasoning Ex 14.3

Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 14 Mathematical Reasoning Ex 14.3 Textbook Questions and Answers.

Gujarat Board Textbook Solutions 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:

1. All rational numbers are real and all numbers are not complex.
2. Square of an integer is positive or negative.
3. The sand heats up quickly in the sun and does not cool down fast at night.
4. x = 2 and x = 3 are the roots of the equation 3x² – x – 10 = 0.

Solution:
1. Connecting word ‘AND’.
p : All rational numbers are real.
q : All numbers are not complex.

2. Connecting word ‘OR’.
p : The square of an integer is positive.
q : The square of an integer is negative.

3. Connecting word ‘AND’.
p : The sand heats up quickly in the sun.
q : The sand does not cool down fast at night.

4. Connecting word ‘AND’.
p : x = 2 is the root of equation 3x² – x – 10 = 0.
q : x = 3 is the root of equation 3x² – x – 10 = 0.

Question 2.
Identify the quantifier in the following statements and write negation of the statement.

1. There exists a number which is equal to its square.
2. For every real number x, x is less than x + 1.
3. There exists a capital for every state of India.

Solution:
1. 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.

2. 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.

3. 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 your answers?

1. x + y = y + x is true for every real numbers x and y.
2. There exists real numbers x and y for which x + y = y + x.

Solution:
Statement (i) and (ii) are not the negation of each other.

Question 4.
State whether the “OR” used in the following statements is “exclusive” or inclusive. Give reasons for your answers.

1. Sun rises or Moon sets.
2. To apply for a driving licence, you should have a ration card or passport.
3. All integers are positive or negative.

Solution:
1. When sun rise, the moon sets. One of the happenings will take place.
Here, ‘OR’ is exclusive.

2. To apply for a driving licence either a ration card or a passport or both can be used.
“OR” used here is inclusive.

3. All integers are positive or negative.
An integers cannot be both + ve and – ve at a time.
∴ Here OR is exclusive.