Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic 1.2 Solutions

Question 1. Express the following statements in symbolic form:
(i) e is a vowel or 2 + 3 = 5.
(ii) Mango is a fruit but potato is a vegetable.
Answer:
(i) Let p : e is a vowel.
q : \( 2 + 3 = 5 \).
Then the symbolic form of the given statement is \( p \vee q \).

(ii) Let p : Mango is a fruit.
q : Potato is a vegetable. Here, the word 'but' acts as a conjunction, which is represented by the logical 'and' operator.
Then the symbolic form of the given statement is \( p \wedge q \).
In simple words: We can write everyday sentences using math symbols by replacing statements with letters like p and q, and connecting words like 'or' with \( \vee \) and 'but' or 'and' with \( \wedge \).

🎯 Exam Tip: Always clearly define your statements p and q before writing the final symbolic form, and remember that 'but' is logically equivalent to 'and' (\( \wedge \)).

Question 1. Express the following statements in symbolic form:
(iii) Milk is white or grass is green.
(iv) I like playing but not singing.
(v) Even though it is cloudy, it is still raining.
Answer:
(iii) Let \( p \) : Milk is white.
\( q \) : Grass is green.
Then the symbolic form of the given statement is \( p \vee q \).

(iv) Let \( p \) : I like playing.
\( q \) : I am not singing.
Then the symbolic form of the given statement is \( p \wedge q \).

(v) The given statement is equivalent to: It is cloudy and it is still raining.
Let \( p \) : It is cloudy.
\( q \) : It is still raining.
Then the symbolic form of the given statement is \( p \wedge q \). Understanding how English connectives translate to mathematical symbols is key to logic.
In simple words: We replace simple sentences with letters like \( p \) and \( q \), and use symbols like \( \wedge \) for 'and' or 'but', and \( \vee \) for 'or'.

🎯 Exam Tip: Words like 'but', 'yet', and 'even though' always translate to the conjunction operator \( \wedge \) in symbolic logic.

Question 2. Write the truth values of the following statements:
(i) Earth is a planet and Moon is a star.
(ii) 16 is an even number and 8 is a perfect square.
Answer:
(i) Let \( p \) : Earth is a planet.
\( q \) : Moon is a star.
Then the symbolic form of the given statement is \( p \wedge q \).
The truth values of \( p \) and \( q \) are T and F respectively.
\( \therefore \) the truth value of \( p \wedge q \) is F. [\( \text{T} \wedge \text{F} \equiv \text{F} \)]

(ii) Let \( p \) : 16 is an even number.
\( q \) : 8 is a perfect square.
Then the symbolic form of the given statement is \( p \wedge q \).
The truth values of \( p \) and \( q \) are T and F respectively.
\( \therefore \) the truth value of \( p \wedge q \) is F. [\( \text{T} \wedge \text{F} \equiv \text{F} \)]
Determining the individual truth value of each component statement is the first step to finding the overall truth value.
In simple words: For an 'and' statement to be true, both parts must be true. Since the Moon is not a star and 8 is not a perfect square, both compound statements are false.

🎯 Exam Tip: Always write down the individual truth values of \( p \) and \( q \) clearly before evaluating the truth value of the combined symbolic statement.

Question (ii) 16 is an even number and 8 is a perfect square.
Answer:
Let \( p \) : 16 is an even number.
\( q \) : 8 is a perfect square.
Then the symbolic form of the given statement is \( p \wedge q \).
The truth values of \( p \) and \( q \) are T and F respectively.

\( \implies \) the truth value of \( p \wedge q \) is F. [\( T \wedge F \equiv F \)]
In simple words: While 16 is indeed an even number, 8 is not a perfect square. Since the word "and" requires both parts to be true, the combined statement is false.

🎯 Exam Tip: Remember that "and" (\( \wedge \)) requires both component statements to be true for the compound statement to be true.

Question (iii) A quadratic equation has two distinct roots or 6 has three prime factors.
Answer:
Let \( p \) : A quadratic equation has two distinct roots.
\( q \) : 6 has three prime factors.
Then the symbolic form of the given statement is \( p \vee q \).
The truth values of both \( p \) and \( q \) are F.

\( \implies \) the truth value of \( p \vee q \) is F. [\( F \vee F \equiv F \)]
In simple words: Both parts of the statement are false. Since the statement uses "or", it is only true if at least one part is true. Since both are false, the whole statement is false.

🎯 Exam Tip: For "or" (\( \vee \)) statements, the compound statement is only false when both individual statements are false.

Question (iv) The Himalayas are the highest mountains but they are part of India in the northeast.
Answer:
Let \( p \) : The Himalayas are the highest mountains.
\( q \) : They are part of India in the northeast.
Then the symbolic form of the given statement is \( p \wedge q \).
The truth values of both \( p \) and \( q \) are T.

\( \implies \) the truth value of \( p \wedge q \) is T. [\( T \wedge T \equiv T \)]
In simple words: Both parts of the statement are true. Since they are joined by "but" (which works like "and"), the entire statement is true.

🎯 Exam Tip: Words like "but", "yet", "still", and "though" are translated using the conjunction operator (\( \wedge \)) in symbolic logic.