GSEB Class 11 Maths Solutions Chapter 14 Mathematical Reasoning Exercise 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 \( 3x^2 - x - 10 = 0 \).
Answer:
1. The connecting word is 'AND'.
p: All rational numbers are real.
q: All numbers are not complex.

2. The connecting word is 'OR'.
p: The square of an integer is positive.
q: The square of an integer is negative.

3. The connecting word here is 'AND'.
p: Sand warms up quickly in the sun.
q: Sand does not cool down quickly at night.

4. The connecting word is 'AND'.
p: \( x = 2 \) is a root of the equation \( 3x^2 - x - 10 = 0 \).
q: \( x = 3 \) is a root of the equation \( 3x^2 - x - 10 = 0 \).
In simple words: For each sentence, first find the word that joins two ideas (like 'AND' or 'OR'). Then, separate those two ideas into their own simple statements, labeling them 'p' and 'q'.

Exam Tip: Always clearly identify the connecting word and then separate the compound statement into its individual simple statements, labeling them 'p' and 'q'.

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.
Answer:
1. The quantifier present in this statement is 'There exists'.
p: There exists a number which is equal to its square.
\( \sim p \): There does not exist a number which is equal to its square.

2. The quantifier here is 'For every'.
p: For every real number \( x \), \( x < x + 1 \).
\( \sim p \): There exists a real number \( x \) such that \( x \geq x + 1 \).

3. The quantifier identified here is 'There exists'.
p: There exists a capital for every state of India.
\( \sim p \): There does not exist a capital for every state of India.
In simple words: Find the special phrase that shows how many things the statement applies to (like 'There exists' or 'For every'). Then, write the opposite of the original statement, making sure to negate both the quantifier and the idea.

Exam Tip: Remember that negating 'For every' changes it to 'There exists', and negating 'There exists' changes it to 'For every'. Always negate the condition as well.

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 \( x + y = y + X \).
Answer: The provided pair of statements are not negations of each other. Statement 1 mentions that \( x + y = y + x \) holds true for all real numbers \( x \) and \( y \). This represents a universal mathematical property. Statement 2 suggests that at least one pair of real numbers \( x \) and \( y \) exists for which \( x + y = y + x \). This assertion is also correct, as the property holds for every pair. For two statements to be negations, if one is true, the other must be false, and conversely. As both statements here are accurate, they cannot serve as negations for one another. The proper negation of Statement 1 would be: 'There exist real numbers \( x \) and \( y \) for which \( x + y \neq y + x \)'.
In simple words: These two statements are not opposites. The first says a math rule is always true for all numbers. The second says that there's at least one time that rule is true. Both are actually true, so they can't be negations of each other, because negations mean one is true and the other is false.

Exam Tip: For two statements to be negations, they must have opposite truth values. If both are true or both are false under the same conditions, they are not negations.

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.
Answer:
1. When the sun rises, the moon sets. Only one of these events will occur at any given moment. Therefore, the 'OR' used here is exclusive.

2. To submit an application for a driving license, a person can use either a ration card or a passport, or even both documents. This indicates that the 'OR' employed here is inclusive.

3. All integers are considered positive or negative. An integer cannot simultaneously be both positive and negative. Thus, the 'OR' in this statement is exclusive.
In simple words: Decide if the 'OR' means you can pick only one option (exclusive) or if you can pick one, the other, or both options (inclusive). Explain why you chose that type of 'OR' for each sentence.

Exam Tip: An 'exclusive OR' means only one of the conditions can be true, never both. An 'inclusive OR' means one, the other, or both conditions can be true.