OP Malhotra Class 11 Maths Solutions Chapter 27 Mathematical Reasoning Exercise 27 (C)

Question 1. Write the given statement in symbolic form using the letter in parentheses to represent the corresponding component.
(i) This is April (p) and income tax returns must be filed (q).
(ii) Accountancy is a required subject for Chartered Accountants (m) but not for engineers (n).
(iii) Mukesh Patel is a teacher (t) or a lawyer (u).
(iv) Jack went up the hill (c) and Jill went up the hill (d).
(v) I plan to take science (a) or commerce (c) in class 11.
(vi) I will not drive to Jaipur (~ d) but I shall go by train (i) or by plane (p).
Answer:
(i) The statement "This is April and income tax returns must be filed" can be written as \( p \wedge q \). The symbol \( \wedge \) means "and".
(ii) The statement "Accountancy is a required subject for Chartered Accountants but not for engineers" is \( m \wedge \sim n \). Here, \( \sim n \) means "not for engineers".
(iii) The statement "Mukesh Patel is a teacher or a lawyer" is \( t \vee u \). The symbol \( \vee \) means "or".
(iv) The statement "Jack went up the hill and Jill went up the hill" is \( c \wedge d \). This shows two events happening together.
(v) The statement "I plan to take science or commerce in class 11" is a disjunction of 'a' and 'c', written as \( a \vee c \). This means one of the options will be chosen.
(vi) The statement "I will not drive to Jaipur but I shall go by train or by plane" is \( (\sim d) \wedge (i \vee p) \). This combines a negation with an "or" choice, linked by "but".
In simple words: We use symbols like \( \wedge \) for "and", \( \vee \) for "or", and \( \sim \) for "not" to turn sentences into shorter logical expressions. Each letter represents a simple part of the sentence.

🎯 Exam Tip: Pay close attention to keywords like "and", "or", "not", and "but". "But" often acts like "and" in logic, showing two ideas are both true even if one is a contrast. Remember to correctly identify the components before assigning symbols.

Question 2. Let p be "Shruti can type”, and let q be “Shruti takes shorthand.” Write the following statements in symbolic form :
(i) Shruti can type and take shorthand.
(ii) Shruti can type but she does not take shorthand.
(iii) Shruti can neither type nor take shorthand.
(iv) It is not true that Shruti can type and take shorthand.
Answer:
Given statements:
\( p \): Shruti can type
\( q \): Shruti takes shorthand

(i) The compound statement "Shruti can type and take shorthand" is a conjunction of \( p \) and \( q \). This means both actions happen. So, its symbolic form is \( p \wedge q \).
(ii) The statement "Shruti can type but she does not take shorthand" is a conjunction of \( p \) and \( \sim q \). This means Shruti does one thing but not the other. So, its symbolic form is \( p \wedge \sim q \).
(iii) The statement "Shruti can neither type nor take shorthand" means Shruti does not type AND she does not take shorthand. This is a conjunction of \( \sim p \) and \( \sim q \). So, its symbolic form is \( \sim p \wedge \sim q \).
(iv) The statement "It is not true that Shruti can type and take shorthand" is the negation of the conjunction of \( p \) and \( q \). This negates the entire combined action. So, its symbolic form is \( \sim (p \wedge q) \).
In simple words: When we are given simple statements like 'p' and 'q', we can use 'not' (\( \sim \)), 'and' (\( \wedge \)), and 'or' (\( \vee \)) to write more complex sentences in a short, symbolic way. This helps in understanding the logical structure clearly.

🎯 Exam Tip: Break down complex sentences into their simplest parts, identify the main connectors ("and", "or", "not"), and then apply the corresponding logical symbols. Be careful with "neither...nor...", which translates to "not...and not...".

Question 3. Use p : Ramesh is rich ; q : Pradeep is poor. Think of “poor” as “not rich”, and write each of these statements in symbolic form.
(i) Ramesh is poor and Pradeep is rich.
(ii) Pradeep and Ramesh are both rich.
(iii) Neither Ramesh nor Pradeep is rich.
(iv) Ramesh is not rich and Pradeep is poor.
(v) It is not true that Ramesh and Pradeep both are rich.
(vi) Either Ramesh is poor or Pradeep is poor.
(vii) Either Ramesh or Pradeep is rich.
Answer:
Given statements:
\( p \): Ramesh is rich
\( q \): Pradeep is poor
Based on the definitions:
\( \sim p \): Ramesh is not rich (or Ramesh is poor)
\( \sim q \): Pradeep is not poor (or Pradeep is rich)

(i) "Ramesh is poor and Pradeep is rich" is represented as \( \sim p \wedge \sim q \). This combines Ramesh being poor with Pradeep being rich.
(ii) "Pradeep and Ramesh are both rich" is represented as \( \sim q \wedge p \). This states that both individuals have the quality of being rich.
(iii) "Neither Ramesh nor Pradeep is rich" means Ramesh is not rich AND Pradeep is not rich. This is \( \sim p \wedge q \). Here, \( q \) already means "Pradeep is poor", which implies he is not rich.
(iv) "Ramesh is not rich and Pradeep is poor" is represented as \( \sim p \wedge q \). This directly combines the two simple statements as defined.
(v) "It is not true that Ramesh and Pradeep both are rich" is the negation of "Ramesh is rich and Pradeep is rich". This is \( \sim (p \wedge \sim q) \). We negate the entire combined idea.
(vi) "Either Ramesh is poor or Pradeep is poor" is represented as \( \sim p \vee q \). This offers two possibilities for their financial status.
(vii) "Either Ramesh or Pradeep is rich" is represented as \( p \vee \sim q \). This implies that at least one of them possesses wealth.
In simple words: We translate sentences about people's wealth into logical symbols. We use 'p' for Ramesh being rich and 'q' for Pradeep being poor. Then, we use 'not' (\( \sim \)), 'and' (\( \wedge \)), or 'or' (\( \vee \)) to build the symbolic form of each sentence.

🎯 Exam Tip: Carefully define the negation of each simple statement before combining them. Remember that "not rich" is equivalent to "poor" in this context, and "not poor" is equivalent to "rich". This makes the symbolic translation more accurate.

Question 4. Use p: I like this school ; q : I like Mr. Sexena. Express each of the following statements in words.
(i) \( p \wedge q \)
(ii) \( \sim q \)
(iii) \( \sim p \)
(iv) \( (\sim p) \wedge (\sim q) \)
(v) \( (\sim p) \wedge q \)
(vi) \( p \vee q \)
(vii) \( \sim (p \wedge q) \)
(viii) \( \sim [(\sim p) \wedge q] \)
Answer:
Given statements:
\( p \): I like this school
\( q \): I like Mr. Sexena

(i) \( p \wedge q \) means "I like this school and I like Mr. Sexena." This is a conjunction, showing both things are true.
(ii) \( \sim q \) means "I do not like Mr. Sexena." This is the negation of statement \( q \).
(iii) \( \sim p \) means "I do not like this school." This is the negation of statement \( p \).
(iv) \( (\sim p) \wedge (\sim q) \) means "I do not like this school and I do not like Mr. Sexena." This combines the negations of both original statements.
(v) \( (\sim p) \wedge q \) means "I do not like this school but I like Mr. Sexena." The "but" acts like "and" here, indicating both parts are true.
(vi) \( p \vee q \) means "I like this school or I like Mr. Sexena." This is a disjunction, indicating at least one of the statements is true.
(vii) \( \sim (p \wedge q) \) means "It is not true that I like this school and I like Mr. Sexena." This negates the entire idea of liking both.
(viii) \( \sim [(\sim p) \wedge q] \) means "It is not true that I do not like this school and I like Mr. Sexena." This negates the combination of not liking the school but liking Mr. Sexena.
In simple words: We take short logical symbols and turn them back into full English sentences. \( \wedge \) means "and", \( \vee \) means "or", and \( \sim \) means "not". When \( \sim \) is outside brackets, it means "It is not true that..." for the whole part inside.

🎯 Exam Tip: When converting symbolic forms to words, carefully observe the scope of the negation symbol (\( \sim \)). If it's applied to a single variable, it negates that specific statement. If it's applied to a compound statement in parentheses, it means "It is not true that..." for the entire compound statement.

Question 5. Give the negation of each of the following statements.
(i) Either he is bald or he is tall.
(ii) Nobody does not like Madhuri.
(iii) It is not true that the set of prime numbers is finite.
(iv) All circles are round.
(v) Some students passed this course.
Answer:
(i) Statement: "Either he is bald or he is tall."
Let \( p \): he is bald, \( q \): he is tall. The statement is \( p \vee q \).
Its negation is \( \sim (p \vee q) \), which means \( \sim p \wedge \sim q \) (De Morgan's Law).
So, the negation in words is: "He is not bald and he is not tall." This means he lacks both qualities.
(ii) Statement: "Nobody does not like Madhuri." This sentence is actually double negative, meaning "Everybody likes Madhuri."
The negation of "Everybody likes Madhuri" is "Somebody does not like Madhuri." This introduces an exception to the universal liking.
(iii) Statement: "It is not true that the set of prime numbers is finite." This implies the set of prime numbers is infinite.
The negation is: "The set of prime numbers is finite." We simply remove the "not true" part to reverse the meaning.
(iv) Statement: "All circles are round." This is a universal statement.
The negation is: "All circles are not round" or "It is not the case that all circles are round." Alternatively, "Some circles are not round." This means there is at least one circle that is not round.
(v) Statement: "Some students passed this course." This is an existential statement.
The negation is: "No student passed this course" or "It is not true that some students passed this course." This implies that every student failed or did not pass the course.
In simple words: To negate a statement, you reverse its truth value. If it says "all," the negation might say "some are not." If it says "some," the negation might say "none." If it says "or," the negation uses "and" with "not" for each part, and vice versa.

🎯 Exam Tip: Remember De Morgan's laws for negating compound statements: \( \sim (p \wedge q) \equiv \sim p \vee \sim q \) and \( \sim (p \vee q) \equiv \sim p \wedge \sim q \). For "all" statements, the negation often starts with "some are not". For "some" statements, the negation is "none".