Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic 1.7 Solutions

Question 1. Write the dual of each of the following:
(i) \( (p \vee q) \vee r \)
(ii) \( \sim(p \vee q) \wedge [p \vee \sim(q \wedge \sim r)] \)
Answer:
(i) \( (p \wedge q) \wedge r \)
(ii) \( \sim(p \wedge q) \vee [p \wedge \sim(q \vee \sim r)] \)
To find the dual of a statement, we replace the disjunction operator with the conjunction operator and vice versa.
In simple words: To find the dual of a logical statement, you simply swap all OR symbols (\( \vee \)) with AND symbols (\( \wedge \)), and vice versa, while keeping the negations exactly the same.

🎯 Exam Tip: Remember that finding a dual only involves swapping \( \vee \) with \( \wedge \) (and T with F if present). Do not change the negation (\( \sim \)) symbols in the statement.

Question 1. Write the dual of the following statement patterns:
(iii) \( p \vee (q \vee r) \equiv (p \vee q) \vee r \)
(iv) \( \sim(p \wedge q) \equiv \sim p \vee \sim q \)
Answer:
(iii) \( p \wedge (q \wedge r) \equiv (p \wedge q) \wedge r \)
(iv) \( \sim(p \vee q) \equiv \sim p \wedge \sim q \) This process of swapping operators helps us find dual statements in mathematical logic.
In simple words: To find the dual of a logical expression, we simply swap the 'OR' symbol (\( \vee \)) with the 'AND' symbol (\( \wedge \)) and keep everything else the same.

🎯 Exam Tip: Remember that finding a dual only involves swapping \( \vee \) with \( \wedge \) (and vice versa) and \( T \) with \( F \) (and vice versa); do not change the negation (\( \sim \)) symbols.

Question 2. Write the dual statement of each of the following compound statements:
(i) 13 is prime number and India is a democratic country.
(ii) Karina is very good or everybody likes her.
(iii) Radha and Sushmita can not read Urdu.
(iv) A number is real number and the square of the number is non-negative.
Answer:
(i) 13 is prime number or India is a democratic country.
(ii) Karina is very good and everybody likes her.
(iii) Radha or Sushmita can not read Urdu.
(iv) A number is real number or the square of the number is non-negative. These dual statements are formed by replacing the connective 'and' with 'or', and vice versa, without changing the truth values of the individual simple statements.
In simple words: To write the dual of a word statement, we change the word 'and' to 'or', and change 'or' to 'and'.

🎯 Exam Tip: When writing duals for verbal statements, only change the connectives 'and'/'or'. Do not change the negation of the statements (i.e., do not change 'is' to 'is not').