Interactive ISC Mathematics Class 11 OP Malhotra Solutions Chapter 27 Mathematical Reasoning Ex 27(f) engage students in active learning and exploration.
S Chand Class 11 ICSE Maths Solutions Chapter 27 Mathematical Reasoning Ex 27(f)
Question 1.
Write the converse, inverse and contra-positive of the following statements.
(i) If you do not drink your milk, you will not be strong.
(ii) If you drink milk, you will be strong.
(iii) You will be strong only if you drink your milk.
Solution:
(i) Converse : q ⇒ p : If you are not strong then you do not drink your milk.
Inverse : ~ p ⇒ ~ q : If you drink your milk then you will be strong.
Contrapositive : ~ q ⇒ ~ p : If you are strong then you drink your milk.
(ii) Converse : If you are strong then you drink your milk.
Inverse : If you do not drink your milk then you are not strong.
Contrapositive : If you are not strong then you do not drink your milk.
(iii) Converse (q ⇒ p): If you drink your milk then you are strong.
Inverse (~ p ⇒ ~ q) : If you are strong then you do not drink your milk.
Contrapositive (~ q ⇒ ~ p) : If you do not drink your milk then you are not strong.
Question 2.
Write the converse of each of the following statements. In which cases is the converse true?
(i) If an integer is even, then its square is divisible by 4.
(ii) If it is raining, then there are clouds in the sky.
(iii) In order to get this job, I must be a graduate.
(iv) If Mr. Sexena is elected to office, then all our problems are over.
Solution:
(i) Converse of given statement is q ⇒ p
If square of an integer is divisible by 4 then the integer is even (True)
(ii) Let p : it is raining
q : There are clouds in the sky
Then converse of p ⇒ q be q ⇒ p
i.e. If there are clouds in the sky then it is raining (False)
(iii) Let p : I get this job
q : I must be a graduate Then the converse of p ⇒ q is q ⇒ p
i.e. If I am a graduate then I can get this job, which is false.
(iv) Let p : Mr. Sexena is elected to office
q : all our problems are over
Then converse of p ⇒ q is q ⇒ p
i.e. If all our problems are over then Mr. Sexena is elected to office (False)
Question 3.
Consider the statements :
p : You will work hard
q : You will become wealthy.
Translate each of the symbolic statements into an English sentence.
(i) P ⇒ q
(ii) q ⇒ p
(iii) (~p) ⇒ (~ q)
(iv) (~q) ⇒ (~p)
Solution:
Given statements are ;
p : You will work hard
q : You will become wealthy
(i) p ⇒ q means if you will work hard then you will become wealthy.
(ii) q ⇒ p: If you will become wealthy then you will work hard
(iii) ~ p ⇒ ~ q : If you will not weak hard then you will not become wealthy
(iv) ~ q ⇒ ~ p : If you will not become wealthy then you will not work hard.
Question 4.
Compare the following statements :
(i) P ⇒ q
(ii) If p, then q
(iii) p is a sufficient condition for q.
(iv) q is a necessary condition for p.
(v) p, only if q.
Solution:
All the given five statements are equivalent and all are equivalent to p ⇒ q.
Question 5.
Construct truth tables for each of the following :
(i) (p ⇒ q) ∧ (q ⇒ p)
(ii) q ⇒ [(~p) q]
(iii) [(~p) ∧ q] ⇒ (p ∨ q)
Solution:
(i) The truth table for (p ⇒ q) ∧ (q ⇒ p) is given are under:
| p | q | P ⇒ q | q ⇒ p | (p ⇒ q) ∧ (q ⇒ p) |
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |
(ii) The truth tale for q ⇒ [(~ p) ∨ q] is given as under:
| s | q | ~ p | ~ p ∨ q | q ⇒ [(~p) ∨ q] |
| T | T | F | T | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | F | T | T | T |
(iii) The truth table for [(~ p) ∧ q] ⇒ (p ∨ q) is given as under :
| p | q | ~ p | ~ p ∧ q | p ∨ q | [~p ∧ q] ⇒ p ∨ q |
| T | T | F | F | T | T |
| T | F | F | F | T | T |
| F | T | T | T | T | T |
| F | F | T | F | F | T |
| T | T | F | F | T | T |
Question 6.
Write the converse, inverse and contra-positive for the statement (~ p) ⇒ q.
Solution:
Given statement is ~p ⇒ q
Converse :q ⇒ ~p
Inverse : ~ (~p) ⇒ ~ q i.e. p ⇒ ~ q
Contrapositive : ~ q ⇒ ~ (~p) i.e. ~ q ⇒ p
Question 7.
Write the inverse of the converse of p ⇒ q.
Solution:
Given statement be p ⇒ q
∴ its converse : q ⇒ p
Thus required inverse be ~q ⇒ ~p
Question 8.
Write the converse of the inverse of p ⇒ q.
Solution:
The inverse of p ⇒ q be ~p ⇒ ~ q
Then its converse be ~ q ⇒ ~ p
Question 9.
Write the contrapositive of the inverse of p ⇒ q.
Solution:
Given statement be p ⇒ q
Inverse : ~ p ⇒ ~q
Contrapositive : ~ (~ q) ⇒ ~ (~ p)
i.e. q ⇒ p
Question 10.
Write the converse of the contrapositive of p ⇒ q.
Solution:
The contrapositive of p ⇒ q be ~q ⇒ ~p
∴ its converse be ~ p ⇒ ~ q
Question 11.
Write the contrapositive of the contrapositive of p ⇒ q.
Solution:
The contrapositive of p ⇒ q be ~ q ⇒ ~p
Then the contrapositive of ~ q
⇒ ~p be ~ (~p) ⇒ ~ (~ q) i.e. p ⇒ q.
Question 12.
Does completing each of the problems 6 through 10 result in a conditional? What is the relationship of each resulting condition to the original conditional P ⇒ q?
Solution:
Yes, contrapositive (# 7) ; contrapositive (# 8); converse (# 9); Increase (# 10) and original condition (#11)
Question 13.
If p and q are any two propositions then prepare the truth table for p ⇒ q, ~q ⇒ ~p and show that the above statements are equivalent.
Hence, or otherwise determine which of the following two arguments is valid?
(i) Given : If you work hard, then you pass the course.
Given : You did not work hard.
Conclusion : You did not pass the course.
(ii) Given : If you work hard, then you pass the course.
Given : You did not pass the course.
Conclusion : You did not work hard.
Solution:
The truth table is given as under :
| I | II | III | IV | V | VI | VII | VIII |
| p | q | p ⇒ q | q ⇒ p | ~P | ~q | ~p ⇒ ~q | ~q ⇒ ~p |
| T | T | T | T | F | F | T | T |
| T | F | F | T | F | T | T | F |
| F | T | T | F | T | F | F | T |
| F | F | T | T | T | T | T | T |
From column III and VIII; we have p ⇒ q ≡ ~q ⇒ ~p
From column IVth and VII; we have q ⇒ p ≡ ~p ⇒ ~q
(i) Let p : You work hard, q : You pass the course,
~ p : You did not work hard, ~ q : You did not pass the course
Now p ⇒ q and ~p ⇒ ~q are not equivalent statement
∴ p ⇒ q Then ~p ⇒ ~ q is not valid.
(ii) p ⇒ q and ~q ⇒ ~p are equivalent statements.
Thus, p ⇒ q then ~ q ⇒ ~ p is valid.