Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic 1.8 Solutions

Question 1. Write the negation of each of the following statements:
(i) All the stars are shining if it is night.
Answer:
The given statement can be written as:
If it is night, then all the stars are shining.
Let \( p \) : It is night.
\( q \) : All the stars are shining.
Then the symbolic form of the given statement is \( p \rightarrow q \). Understanding the conditional structure is key to finding its correct negation.
Its negation is:
\( \implies \sim(p \rightarrow q) \equiv p \wedge \sim q \)
Therefore, the negation of the given statement is: It is night and some stars are not shining.
In simple words: To negate an "if-then" statement, you keep the first condition exactly the same and combine it with the opposite of the second condition using "and".

🎯 Exam Tip: Always rewrite the conditional statement in the standard "If p, then q" form first. This makes it much easier to correctly apply the negation rule \( p \wedge \sim q \) without making mistakes.

Question 1. Write the negation of the following:
(i) If it is night, then all the stars are shining.
(ii) \( \forall n \in N, n + 1 > 0 \)
(iii) \( \exists n \in N \), such that \( (n^2 + 2) \) is odd number.
(iv) Some continuous functions are differentiable.
Answer:
(i) Since, \( \sim(p \rightarrow q) \equiv p \wedge \sim q \), the negation of the given statement is 'It is night and all the stars are not shining.'
(ii) The negation of the given statement is '\( \exists n \in N \), such that \( n + 1 \le 0 \).'
(iii) The negation of the given statement is '\( \forall n \in N, n^2 + 2 \) is not an odd number.'
(iv) The negation of a given statement is 'All continuous functions are not differentiable.'
In simple words: To write the negation of a statement, we change universal quantifiers like 'for all' (\( \forall \)) to existential quantifiers like 'there exists' (\( \exists \)), and flip the inequality or condition to its opposite.

🎯 Exam Tip: Remember that the negation of \( \forall \) (for all) is always \( \exists \) (there exists) and vice versa, and make sure to change the inequality sign (e.g., \( > \) becomes \( \le \)).

Question 2. Using the rules of negation, write the negations of the following:
(i) \( (p \rightarrow r) \wedge q \)
(ii) \( \sim(p \vee q) \rightarrow r \)
(iii) \( (\sim p \wedge q) \wedge (\sim q \vee \sim r) \)
Answer:
(i) The negation of \( (p \rightarrow r) \wedge q \) is:
\( \sim[(p \rightarrow r) \wedge q] \equiv \sim(p \rightarrow r) \vee (\sim q) \) .....[Negation of conjunction]
\( \equiv (p \wedge \sim r) \vee (\sim q) \) ......[Negation of implication]

(ii) The negation of \( \sim(p \vee q) \rightarrow r \) is:
\( \sim[\sim(p \vee q) \rightarrow r] \equiv \sim(p \vee q) \wedge (\sim r) \) .....[Negation of implication]
\( \equiv (\sim p \wedge \sim q) \wedge (\sim r) \) ......[Negation of disjunction]

(iii) The negation of \( (\sim p \wedge q) \wedge (\sim q \vee \sim r) \) is:
\( \sim[(\sim p \wedge q) \wedge (\sim q \vee \sim r)] \equiv \sim(\sim p \wedge q) \vee \sim(\sim q \vee \sim r) \) ......[Negation of conjunction]
In simple words: To find the negation of compound statements, apply the rules step-by-step: the negation of 'and' (\( \wedge \)) becomes 'or' (\( \vee \)), and the negation of an implication \( p \rightarrow q \) becomes \( p \wedge \sim q \).

🎯 Exam Tip: Always write the name of the rule (like 'Negation of conjunction' or 'Negation of implication') next to each step to secure full marks in board exams.

p>Question 3. Write the converse, inverse, and contrapositive of the following statements:
(i) If it snows, then they do not drive the car.
(ii) If he studies, then he will go to college.
Answer:
(i) Let \( p \) : It snows.
\( q \) : They do not drive the car.
Then the symbolic form of the given statement is \( p \rightarrow q \).
Converse: \( q \rightarrow p \) is the converse of \( p \rightarrow q \).
i.e. If they do not drive the car, then it snows.
Inverse: \( \sim p \rightarrow \sim q \) is the inverse of \( p \rightarrow q \).
i.e. If it does not snow, then they drive the car.
Contrapositive: \( \sim q \rightarrow \sim p \) is the contrapositive of \( p \rightarrow q \).
i.e. If they drive the car, then it does not snow.

(ii) Let \( p \) : He studies.
\( q \) : He will go to college.
Then the symbolic form of the given statement is \( p \rightarrow q \).
Converse: \( q \rightarrow p \) is the converse of \( p \rightarrow q \).
i.e. If he will go to college, then he studies.
Inverse: \( \sim p \rightarrow \sim q \) is the inverse of \( p \rightarrow q \).
i.e. If he does not study, then he will not go to college.
Contrapositive: \( \sim q \rightarrow \sim p \) is the contrapositive of \( p \rightarrow q \).
i.e. If he will not go to college, then he does not study. These logical transformations help in analyzing conditional statements in mathematical reasoning.
In simple words: For any "If-Then" statement, the converse swaps the two parts, the inverse negates both parts, and the contrapositive both swaps and negates them. The contrapositive always means the exact same thing as the original statement.

🎯 Exam Tip: Remember that a conditional statement and its contrapositive always share the same truth value, while the converse and inverse are also logically equivalent to each other.

Question 4. With proper justification, state the negation of each of the following:
(i) \( (p \rightarrow q) \vee (p \rightarrow r) \)
Answer:
The negation of \( (p \rightarrow q) \vee (p \rightarrow r) \) is:
\( \sim[(p \rightarrow q) \vee (p \rightarrow r)] \equiv \sim(p \rightarrow q) \wedge \sim(p \rightarrow r) \) .....[Negation of disjunction]
\( \equiv (p \wedge \sim q) \wedge (p \wedge \sim r) \) ...[Negation of implication]
This step-by-step simplification clearly demonstrates how compound logical statements can be systematically negated.
In simple words: To find the opposite (negation) of an "OR" statement, we find the opposite of both parts and connect them with "AND". Since the opposite of "If p then q" is "p and not q", we apply this rule to both parts.

🎯 Exam Tip: Always write down the name of the logic rule (like De Morgan's Law or Negation of Implication) next to each step to secure full marks for justification.

Question (ii). \( (p \leftrightarrow q) \vee (\sim q \rightarrow \sim r) \)
Answer:
The negation of \( (p \leftrightarrow q) \vee (\sim q \rightarrow \sim r) \) is:
\( \sim [(p \leftrightarrow q) \vee (\sim q \rightarrow \sim r)] \)
\( \equiv \sim (p \leftrightarrow q) \wedge \sim (\sim q \rightarrow \sim r) \) .....[Negation of disjunction]
\( \equiv [(p \wedge \sim q) \vee (q \wedge \sim p)] \wedge [\sim q \wedge \sim (\sim r)] \) .....[Negation of biconditional and implication]
\( \equiv [(p \wedge \sim q) \vee (q \wedge \sim p)] \wedge (\sim q \wedge r) \) .....[Negation of negation]
This step-by-step simplification helps in breaking down complex logical statements into their simplest negated forms.
In simple words: To negate an 'OR' statement, we negate both parts and change the 'OR' to 'AND', then apply the specific negation rules for conditional and biconditional statements.

🎯 Exam Tip: Remember that the negation of \( p \rightarrow q \) is \( p \wedge \sim q \), and always write down the logical reason for each step to secure full marks.

Question (iii). \( (p \rightarrow q) \wedge r \)
Answer:
The negation of \( (p \rightarrow q) \wedge r \) is:
\( \sim [(p \rightarrow q) \wedge r] \)
\( \equiv \sim (p \rightarrow q) \vee (\sim r) \) .....[Negation of conjunction]
\( \equiv (p \wedge \sim q) \vee (\sim r) \) .....[Negation of implication]
This logical equivalence shows how conjunctions behave under negation.
In simple words: To negate an 'AND' statement, we negate both parts and change the 'AND' to 'OR', then simplify the negation of the implication.

🎯 Exam Tip: Be careful with brackets when negating conjunctions, and clearly state 'Negation of conjunction' and 'Negation of implication' as your reasons.