The availability of step-by-step ISC Mathematics Class 11 OP Malhotra Solutions Chapter 27 Mathematical Reasoning Ex 27(d) can make challenging problems more manageable.
S Chand Class 11 ICSE Maths Solutions Chapter 27 Mathematical Reasoning Ex 27(d)
Question 1.
Let p, q, r and s represent simple statements. Assume that p is false, q is true, r is false, and s is true. Determine the truth value of each statement expressed below :
(i) q ∧ r
(ii) r ∨ p
(iii) p ∧ S
(iv) p ∨ s
(v) ~q
(vi) q ∨ s
(vii) ~ r
(viii) s ∧ q
(ix) r ∧ p
Solution:
Given p is false, q is true, r is false and s is true.
(i) The truth value of statement is false as one of the component statement is false.
(ii) The truth value of statement is false since both component statements are false.
(iii) The truth value of compound statement is false since one of the component statement is false.
(iv) The truth value of compound statement is true since one of the component statement is true.
(v) The truth value of statement is false,
(vi) The truth value of q ∨ s is true since both component statements are true.
(vii) The truth value of compound statement is true.
(viii) The compound statement has truth value is true.
since, both component statements are true.
(ix) The truth value of compound statement is false.
since both component statements are false.
Question 2.
Let a, b, c and d represent simple statements. Assume that a ∧ d is true, b ∧ C is false, and ~ c is false.
(i) What is the truth value of a?
(ii) What is the truth value of d?
(iii) What is truth value of c?
(iv) What is the truth value of b?
Solution:
Given a ∧ d is true ⇒ a and d both statements are true.
and b ∧ C is false. …(1)
Since ~ c is false ⇒ c is true
∴ from (1); b is false (∵ F A T = F)
(i) The truth value of a be true.
(ii) The truth value of d be true.
(iii) The truth value of c be true.
(iv) The truth value of b be false.
Question 3.
Assume that two given statements p and q are both true and indicate whether or not you would expect each of the following statements to be true :
(i) P ∧ q
(ii) p ∨ q
(iii) p ∨ (~q)
(iv) (~p) ∨ (~ q)
Solution:
Given statements p and q are both true.
(i) Thus, the statement p ∧ q = T ∧ I = T is also true.
(ii) The statement p ∨ q is also statement.
(iii) p ∨ (~ q) = T ∨ F = T
(iv) (~p) ∨ (~ q) = F ∨ F = F
Construct truth tables for :
Question 4.
(~p) ∧ q
Solution:
| p | q | ~ p | ~ P ∧ q |
| T | F | F | F |
| T | T | F | F |
| F | T | T | T |
| F | F | T | F |
Question 5.
(~p) ∧ (~q)
Solution:
| P | q | ~ p | ~ q | (~ p) ∧ (~ q) |
| T | T | F | F | F |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |
Question 6.
~(p ∧ q)
Solution:
The truth table for ~ (p ∧ q) is given as under:
| p | q | P ∧ q | ~ (P ∧ q) |
| T | T | T | F |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
Question 7.
p ∧ (~ q)
Solution:
The truth table for statement p ∨ (~ q) is given as under:
| P | q | ~q | P ∨ (~q) |
| T | T | F | T |
| T | F | T | T |
| F | T | F | F |
| F | F | T | T |
Question 8.
~ [p ∨ (~ q)]
Solution:
The truth table for ~[p ∨ (~ q)] is given as under:
| P | q | ~ q | P ∨ (~ q) | ~[p ∨ (~q)] |
| T | T | F | T | F |
| T | F | T | T | F |
| F | T | F | F | T |
| F | F | T | T | F |
Question 9.
~ (~ p ∧ ~ q)
Solution:
The truth table for ~ (~p ∧ ~ q) is given as under:
| P | q | ~ P | ~ q | ( ~ p ∧ ~ q) | ~ (~ p ∧ ~ q) |
| T | T | F | F | F | T |
| T | F | F | T | F | T |
| F | T | T | F | F | T |
| F | F | T | T | T | F |
Question 10.
(p ∧ q) ∨ (~p ∧ q)
Solution:
The truth table for (p ∧ q) ∨ (~ p ∧ q) is given as under:
| p | q | P ∧ q | ~ p | ~ p ∧ q | (p ∧ q) ∨ (~ p ∧ q) |
| T | T | T | F | F | T |
| T | F | F | F | F | F |
| F | T | F | T | T | T |
| F | F | F | T | F | F |
Question 11.
p ∧ (q ∨ r)
Solution:
The truth table for p ∧ (q ∨ r) is given as under:
| p | q | r | q ∧ r | P ∧ (q ∨ r) |
| T | T | T | T | T |
| T | T | F | T | T |
| T | F | T | T | T |
| T | F | F | F | F |
| F | T | T | T | F |
| F | T | F | T | F |
| F | F | T | T | F |
| F | F | F | F | F |
Question 12.
(~p ∧ ~ q) ∨ (p ∧ ~ q)
Solution:
The truth table for (~ p ∧ ~ q) ∨ (p ∧ ~ q) is given as under:
| p | q | ~ p | ~ q | (~ p ∧ ~ q) | P ∧ ~ q | (~ p ∧ ~ q) ∨ (p ∧ ~ q) |
| T | T | F | F | F | F | F |
| T | F | F | T | F | T | T |
| F | T | T | F | F | F | F |
| F | F | T | T | T | F | T |
Question 13.
(p ∨ q) ∨ (r ∧ ~ q)
Solution:
The truth table for (p ∨ q) ∨ (r ∧ ~ q) is given as under:
| p | q | r | ~q | P v q | r ∨ ~ q | (p ∨ q) ∨ (r ∧ ~ q) |
| T | T | T | F | T | F | T |
| T | T | F | F | T | F | T |
| T | F | T | T | T | T | T |
| T | F | F | T | T | F | T |
| F | T | T | F | T | F | T |
| F | T | F | F | T | F | T |
| F | F | T | T | F | T | T |
| F | F | F | T | F | F | F |
Question 14.
Let p be “Ananya is beautiful,” and let q be “Ananya is 165 centimetres tall.”
(i) Under what conditions is the statement, “Ananya is beautiful and 165 centimetres tall.” true?
(ii) Under what conditions is the statement, “Ananya is beautiful and 165 centimetres tall,” false?
(iii) Under what conditions is the statement, “Ananya is beautiful or 165 centimetres tall,” true?
(iv) Under what conditions is the statement, “Ananya is beautiful or 165 centimetres tall,” false?
Solution:
Given p : Ananya is beautiful
q: Ananya is 165 centimetres tall
(i) Only if Ananya is both beautiful and 165 cm tall.
(ii) If Ananya is not beautiful and/or not 165 cm tall.
(iii) If Ananya is beautiful and/or 165 cm tall.
(iv) Only if Ananya is neither beautiful nor 165 cm tall.