Students often turn to ISC Mathematics Class 11 OP Malhotra Solutions Chapter 27 Mathematical Reasoning Ex 27(c) to clarify doubts and improve problem-solving skills.
S Chand Class 11 ICSE Maths Solutions Chapter 27 Mathematical Reasoning Ex 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).
Solution:
(i) p ∧ q
(ii) m ∧ ~ n
(iii) t ∨ u
(iv) c ∧ d
(v) Compound statement is the disjunction of a and c i.e. a ∨ c
(vi) The compound statement is the conjunction of ~ d and (disjunction of t and p) i.e. (~ d) A (t ∨ p)
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.
Solution:
Given p : Shruti can type
q : Shruti takes shorthand
(i) Compound statement is the conjunction of p and q i.e. p ∧ q.
(ii) ~ q : she does not takes shorthand The given statement is the conjunction ofp and ~ q i.e. p ∧ q.
(iii) ~ p : Shruti can’t type
∴ given statement is the conjunction of ~p and ~ q i.e. ~p ∧ ~ q.
(iv) p ∧ q : Shruti can type and takes shorthand
∴ ~(p ∧ q): It is not true that Shruti can type and take shorthand.
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.
Solution:
Given statements are : p : Ramesh is rich q : Pradeep is poor ~p : Ramesh is not rich i.e. poor ~ q : Pradeep is not poor i.e. rich.
(i) (~P) ∧ (~ q)
(ii) (~ q) ∧ p
(iii) (~ p) ∧ q
(iv) ~ p ∧ q
(v) ~ [p ∧ (~ q)]
(vi) ~p ∨ q
(vii) p ∨ (~q)
Question 4.
Use p : I like this school ; q : I like Mr. Sexena. Express each of the following statements in words.
(i) p ∧ q
(ii) ~ q
(iii) ~p
(iv) (~p) ∧ (~ q)
(v) (~p) ∧ q
(vi) p ∨ q
(vii) ~ (p ∧ q)
(viii) ~ [(~ p) ∧ q]
Solution:
Given statements are ;
p : I like this school
q : I like Mr. Sexena.
(i) p ∧ q : I like this school and I like Mr. Sexena.
(ii) ~ q : I do not like Mr. Sexena.
(iii) ~p : I do not like this school.
(iv) ~p ∧ ~ q : I do not like this school and I do not like Mr. Sexena.
(v) ~ p ∧ q : I donot like this school but I like Mr. Sexena.
(vi) p v q : I like this school or I like Mr. Sexena.
(vii) ~ (p ∧ q) : It is not true that I like this school and I like Mr. Sexena.
(viii) ~ [(~ p) ∧ q] : It is not true that I donot like this school and I like Mr. Sexena.
Question 5.
Give the negation of each of the following statements.
(i) Either he is bald or he is tali.
(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.
Solution:
(i) Let p : he is bald
q : he is tall.
∴ p ∨ q: either he is bald or he is tall.
Then ~ (p ∨ q) : He is not bald and he is not tall.
(ii) Negation of given statement is that somebody does not like Madhuri.
(iii) Let p : It is not true that set of prime numbers is finite.
∴ ~p : set of prime numbers is finite.
(iv) p : all circles are sound
~ p : all circles are not round or it is not the case that all circles are round.
(v) p : Some students passed this course
~ p : It is not true that, some students passed this course or No student passed this course.