Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic 1.4 Solutions

Question 1. Write the following statements in symbolic form:
(i) If the triangle is equilateral, then it is equiangular.
(ii) It is not true that ‘i’ is a real number.
Answer:
(i) Let \( p \): Triangle is equilateral.
\( q \): It is equiangular.
Then the symbolic form of the given statement is \( p \rightarrow q \).

(ii) Let \( p \): ‘i’ is a real number.
Then the symbolic form of the given statement is \( \sim p \). Converting these statements into mathematical symbols allows us to analyze their truth values more systematically.
In simple words: We represent sentences with letters like \( p \) and \( q \), and use symbols like \( \rightarrow \) for "if-then" and \( \sim \) for "not" to write them in a short mathematical way.

🎯 Exam Tip: Always define the simple declarative statements clearly as \( p \) and \( q \) without any connectives before writing the final symbolic form.

Question 1(ii). [It is not true that 'i' is a real number.]
Answer: Let \( p \) : 'i' is a real number. Then the symbolic form of the given statement is \( \sim p \). This negation operation flips the truth value of the original statement.
In simple words: The statement says 'i' is not a real number, so we write it as not \( p \), which is \( \sim p \).

🎯 Exam Tip: Remember that the symbol \( \sim \) represents negation, which is equivalent to the word 'not' in English.

Question 1. Express the following statements in symbolic form:
(iii) Even though it is not cloudy, it is still raining.
(iv) Milk is white if and only if the sky is not blue.
(v) Stock prices are high if and only if stocks are rising.
(vi) If Kutub-Minar is in Delhi, then Taj Mahal is in Agra.
Answer:
(iii) Let \( p \) : It is cloudy.
\( q \) : It is still raining.
Then the symbolic form of the given statement is \( \sim p \wedge q \).

(iv) Let \( p \) : Milk is white.
\( q \) : Sky is blue.
Then the symbolic form of the given statement is \( p \leftrightarrow (\sim q) \).

(v) Let \( p \) : Stock prices are high.
\( q \) : stocks are rising.
Then the symbolic form of the given statement is \( p \leftrightarrow q \).

(vi) Let \( p \) : Kutub-Minar is in Delhi.
\( q \) : Taj Mahal is in Agra.
Then the symbolic form of the given statement is \( p \rightarrow q \). These logical representations help in analyzing complex mathematical proofs.
In simple words: We can convert everyday sentences into mathematical logic symbols like \( \wedge \) for 'and', \( \leftrightarrow \) for 'if and only if', and \( \rightarrow \) for 'if-then'.

🎯 Exam Tip: Identify the simple statements first, assign them variables like p and q, and then use the correct logical connectives to write the symbolic form.

Question 2. Find the truth value of each of the following statements:
(i) It is not true that \( 3 - 7i \) is a real number.
Answer:
(i) Let \( p \) : \( 3 - 7i \) be a real number.
Then the symbolic form of the given statement is \( \sim p \).
The truth value of \( p \) is F. This is because complex numbers contain an imaginary part \( i \) which distinguishes them from real numbers.
\( \therefore \) the truth value of \( \sim p \) is T. [\( \sim \text{F} \equiv \text{T} \)]
In simple words: Since \( 3 - 7i \) is a complex number, the statement that it is a real number is false, which makes its negation true.

🎯 Exam Tip: Always state the truth value of the individual statement \( p \) first, then apply the negation rule to find the final truth value of \( \sim p \).

Question (ii). If a joint venture is a temporary partnership, then a discount on purchase is credited to the supplier.
Answer: Let \( p \) : Joint venture is a temporary partnership.
\( q \) : Discount on purchases is credited to the supplier.
Then the symbolic form of the given statement is \( p \rightarrow q \).
The truth values of \( p \) and \( q \) are T and F respectively, as a discount on purchase is actually credited to the joint venture account rather than the supplier.

\( \implies \) the truth value of \( p \rightarrow q \) is F. [\( T \rightarrow F \equiv F \)]
In simple words: An "if-then" statement is only false when the first part is true but the second part is false. Since the second part here is false, the whole statement is false.

🎯 Exam Tip: Clearly define the component statements \( p \) and \( q \) before writing the symbolic form to avoid representation errors.

Question (iii). Every accountant is free to apply his own accounting rules if and only if machinery is an asset.
Answer: Let \( p \) : Every accountant is free to apply his own accounting rules.
\( q \) : Machinery is an asset.
Then the symbolic form of the given statement is \( p \leftrightarrow q \).
The truth values of \( p \) and \( q \) are F and T respectively, since accountants must follow standardized GAAP rules rather than their own personal guidelines.

\( \implies \) the truth value of \( p \leftrightarrow q \) is F. [\( F \leftrightarrow T \equiv F \)]
In simple words: A double implication "if and only if" is only true when both parts have the same truth value. Since one is false and the other is true, the statement is false.

🎯 Exam Tip: Remember that a biconditional statement (\( \leftrightarrow \)) is true only when both component statements have identical truth values (both T or both F).

Question (iv). Neither 27 is a prime number nor divisible by 4.
Answer: Let \( p \) : 27 is a prime number.
\( q \) : 27 is divisible by 4.
Then the symbolic form of the given statement is \( \sim p \wedge \sim q \).
The truth values of both \( p \) and \( q \) are F, as 27 is a composite number divisible by 3 and 9, and it is not divisible by 4.

\( \implies \) the truth value of \( \sim p \wedge \sim q \) is T. [\( \sim F \wedge \sim F \equiv T \wedge T \equiv T \)]
In simple words: The statement says 27 is not prime and not divisible by 4. Since both of these claims are true, the entire statement is true.

🎯 Exam Tip: "Neither... nor..." translates to the conjunction of two negations, which is written symbolically as \( \sim p \wedge \sim q \).

Question (v). 3 is a prime number and an odd number.
Answer: Let \( p \) : 3 be a prime number.
\( q \) : 3 is an odd number.
Then the symbolic form of the given statement is \( p \wedge q \).
The truth values of both \( p \) and \( q \) are T, since 3 has only two factors and cannot be divided evenly by 2.

\( \implies \) the truth value of \( p \wedge q \) is T. [\( T \wedge T \equiv T \)]
In simple words: Since 3 is indeed both a prime number and an odd number, both parts are true, making the combined "and" statement true.

🎯 Exam Tip: For a conjunction (\( \wedge \)) to be true, both individual statements must be true; if even one is false, the whole statement becomes false.

Question 3. If \( p \) and \( q \) are true and \( r \) and \( s \) are false, find the true value of each of the following statements:
(i) \( p \wedge (q \wedge r) \)
(ii) \( (p \rightarrow q) \vee (r \wedge s) \)
(iii) \( \sim[(\sim p \vee s) \wedge (\sim q \wedge r)] \)
(iv) \( (p \rightarrow q) \leftrightarrow \sim(p \vee q) \)

Answer:
(i) \( p \wedge (q \wedge r) \)
Truth values of \( p \) and \( q \) are T and truth values of \( r \) and \( s \) are F. This step-by-step substitution helps us systematically evaluate complex logical expressions.
\( p \wedge (q \wedge r) \equiv \text{T} \wedge (\text{T} \wedge \text{F}) \)
\( \equiv \text{T} \wedge \text{F} \)
\( \equiv \text{F} \)
Hence, the truth value of the given statement is false.

(ii) \( (p \rightarrow q) \vee (r \wedge s) \)
\( (p \rightarrow q) \vee (r \wedge s) \equiv (\text{T} \rightarrow \text{T}) \vee (\text{F} \wedge \text{F}) \)
\( \equiv \text{T} \vee \text{F} \)
\( \equiv \text{T} \)
Hence, the truth value of the given statement is true.

(iii) \( \sim[(\sim p \vee s) \wedge (\sim q \wedge r)] \)
\( \sim[(\sim p \vee s) \wedge (\sim q \wedge r)] \equiv \sim[(\sim \text{T} \vee \text{F}) \wedge (\sim \text{T} \wedge \text{F})] \)
\( \equiv \sim[(\text{F} \vee \text{F}) \wedge (\text{F} \wedge \text{F})] \)
\( \equiv \sim(\text{F} \wedge \text{F}) \)
\( \equiv \sim \text{F} \)
\( \equiv \text{T} \)
Hence, the truth value of the given statement is true.

(iv) \( (p \rightarrow q) \leftrightarrow \sim(p \vee q) \)
\( (p \rightarrow q) \leftrightarrow \sim(p \vee q) = (\text{T} \rightarrow \text{T}) \leftrightarrow \sim(\text{T} \vee \text{T}) \)
\( \equiv \text{T} \leftrightarrow \sim(\text{T}) \)
\( \equiv \text{T} \leftrightarrow \text{F} \)
\( \equiv \text{F} \)
Hence, the truth value of the given statement is false.
In simple words: To find if a combined statement is true or false, we replace each letter with its given value (T or F) and solve it step-by-step. We work from the inside of the brackets outward, just like in normal math rules.

🎯 Exam Tip: Always write down the truth values of individual variables first, and show each step of simplification clearly to secure full marks.

Question 3(v). Find the truth value of \( [(p \vee s) \rightarrow r] \vee [\sim(p \rightarrow q) \vee s] \)
Answer:
Given statement: \( [(p \vee s) \rightarrow r] \vee [\sim(p \rightarrow q) \vee s] \)
Assuming the standard truth values where \( p, q \) are True (\( T \)) and \( r, s \) are False (\( F \)):
\( [(p \vee s) \rightarrow r] \vee \sim[\sim(p \rightarrow q) \vee s] \)
\( \equiv [(T \vee F) \rightarrow F] \vee \sim[\sim(T \rightarrow T) \vee F] \)
\( \equiv (T \rightarrow F) \vee \sim(\sim T \vee F) \)
\( \equiv F \vee \sim(F \vee F) \)
\( \equiv F \vee \sim F \)
\( \equiv F \vee T \)
\( \equiv T \)
Hence, the truth value of the given statement is true.
In simple words: By substituting the truth values of the individual statements into the logical expression and simplifying step-by-step, we find that the final result is True.

🎯 Exam Tip: Remember that an implication \( T \rightarrow F \) is the only case where the conditional statement is False; all other cases are True.

Question 3(vi). Find the truth value of \( \sim[p \vee (r \wedge s)] \wedge \sim[(r \wedge \sim s) \wedge q] \)
Answer:
Given statement: \( \sim[p \vee (r \wedge s)] \wedge \sim[(r \wedge \sim s) \wedge q] \)
Assuming the standard truth values where \( p, q \) are True (\( T \)) and \( r, s \) are False (\( F \)):
\( \sim[p \vee (r \wedge s)] \wedge \sim[(r \wedge \sim s) \wedge q] \)
\( \equiv \sim[T \vee (F \wedge F)] \wedge \sim[(F \wedge \sim F) \wedge T] \)
\( \equiv \sim[T \vee F] \wedge \sim[(F \wedge T) \wedge T] \)
\( \equiv \sim T \wedge \sim(F \wedge T) \)
\( \equiv F \wedge \sim F \)
\( \equiv F \wedge T \)
\( \equiv F \)
Hence, the truth value of the given statement is false.
In simple words: We replace each variable with its truth value and simplify the brackets first, which ultimately leads to a final truth value of False.

🎯 Exam Tip: Carefully apply the negation operator \( \sim \) to the truth values inside the brackets before performing the conjunction or disjunction operations.

Question 4. Assuming that the following statements are true:
p : Sunday is a holiday.
q : Ram does not study on holiday.
Find the truth values of the following statements:
(i) Sunday is not holiday or Ram studies on holiday.
(ii) If Sunday is not a holiday, then Ram studies on holiday.
Answer:
Given that \( p \) and \( q \) are true statements (i.e., \( p \equiv T \), \( q \equiv T \)).

(i) Sunday is not holiday or Ram studies on holiday.
The symbolic form of the statement is \( \sim p \vee \sim q \).
Since \( p \) is \( T \) and \( q \) is \( T \), we have \( \sim p \equiv F \) and \( \sim q \equiv F \).
Therefore, \( \sim p \vee \sim q \equiv F \vee F \equiv F \).
We can represent this in the truth table below:

Hence, the truth value of the given statement is F.

(ii) If Sunday is not a holiday, then Ram studies on holiday.
The symbolic form of the given statement is \( \sim p \rightarrow \sim q \).
Since \( p \) is \( T \) and \( q \) is \( T \), we have \( \sim p \equiv F \) and \( \sim q \equiv F \).
Therefore, \( \sim p \rightarrow \sim q \equiv F \rightarrow F \equiv T \).
Hence, the truth value of the given statement is T.
In simple words: We translate the English sentences into logical symbols using p and q, then substitute their known truth values to find the final truth value of each compound statement.

🎯 Exam Tip: When translating "not" statements, always negate the original variable (e.g., "studies" becomes \( \sim q \) if \( q \) is "does not study").

Question. Sunday is a holiday and Ram studies on holiday.
Answer:
The symbolic form of the given statement is \( p \wedge \sim q \).

Therefore, the truth value of the given statement is F.
In simple words: Since the second part of the statement is false, the combined "and" statement is also false.

🎯 Exam Tip: When constructing truth tables, always write down the truth values of individual components step-by-step to avoid simple calculation errors.

Question 5. If p : He swims. q : Water is warm. Give the verbal statements for the following symbolic statements:
(i) \( p \leftrightarrow \sim q \)
(ii) \( \sim(p \vee q) \)
(iii) \( q \rightarrow p \)
(iv) \( q \wedge \sim p \)
Answer:
(i) He swims if and only if the water is not warm.
(ii) It is not true that he swims or water is warm.
(iii) If water is warm, then he swims.
(iv) The water is warm and he does not swim. These verbal translations help in understanding the logical relationships between the two simple statements.
In simple words: We are translating mathematical logic symbols back into regular English sentences using words like 'if and only if', 'not', 'or', 'if...then', and 'and'.

🎯 Exam Tip: Pay close attention to negation symbols (\( \sim \)) and conditional arrows (\( \rightarrow \)) to ensure your English translation matches the exact logical meaning.