Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic 1.3 Solutions

Question 1. Write the negation of each of the following statements:
(i) All men are animals.
(ii) 3 is a natural number.
Answer:
(i) Some men are not animals.
(ii) 3 is not a natural number. Negating a statement changes its truth value to its exact opposite.
In simple words: Negation means writing the opposite of the given statement. For example, 'all' becomes 'some are not', and 'is' becomes 'is not'.

🎯 Exam Tip: Remember that the negation of 'All A are B' is 'Some A are not B', not 'No A are B'. Be careful with quantifiers to avoid losing marks.

Question 1. Write the negation of the following statements:
(iii) It is false that Nagpur is the capital of Maharashtra.
(iv) \( 2 + 3 \neq 5 \).
Answer:
(iii) Nagpur is the capital of Maharashtra.
(iv) \( 2 + 3 = 5 \). This simple equation shows that the sum of 2 and 3 is indeed equal to 5.
In simple words: To negate a statement, we change its meaning to the opposite. "It is false that" is removed to make it positive, and "not equal to" (\( \neq \)) is changed to "equal to" (\( = \)).

🎯 Exam Tip: When negating a statement containing a negation like 'it is false that', simply remove that phrase to write the correct negation.

Question 2. Write the truth value of the negation of each of the following statements:
(i) \( \sqrt{5} \) is an irrational number.
(ii) London is in England.
(iii) For every \( x \in N \), \( x + 3 < 8 \).
Answer:
(i) Let \( p \) : \( \sqrt{5} \) is an irrational number.
The truth value of \( p \) is T.
Therefore, the truth value of \( \sim p \) is F.

(ii) Let \( p \) : London is in England.
The truth value of \( p \) is T.
Therefore, the truth value of \( \sim p \) is F.

(iii) Let \( p \) : For every \( x \in N \), \( x + 3 < 8 \).
The truth value of \( p \) is F.
Therefore, the truth value of \( \sim p \) is T. This is because some natural numbers like 5 make the inequality false.
In simple words: First, find if the statement is true (T) or false (F). The negation will always have the exact opposite truth value, so a true statement becomes false, and a false statement becomes true.

🎯 Exam Tip: Always define the statement as \( p \) first, state its truth value clearly, and then write the truth value of its negation \( \sim p \) to get full marks.