HSC Maths Chapter 1.1: Mathematical Logic Question Bank Solutions for Arts and Science


Q1) Select and write the most appropriate answer from the given alternatives:

i) Which of the following statement is true

a) 3 + 7 =4 or 3 – 7 = 4

b) If Pune is in Maharashtra, then Hyderabad is in Kerala

c) It is false that 12 is not divisible by 3

d) The square of any odd integer is even.

Answer:-

It is false that 12 is not divisible by 3

ii) Which of the following is not a statement

a) 2+2 =4

b) 2 is the only even prime number

c) Come here

d) Mumbai is not in Maharashtra

Answer:-

Come here

iii) If p is any statement then ( p ? ∼ p) is a

a) Contingency

b) Contradiction

c) Tautology

d) None of these

Answer:-

Tautology

iv) If p and q are two statements then (p → q) ↔ (∼ q → ∼ p) is

a) Contradiction

b) Tautology

c) Neither (i) nor (ii)

d) None of these

Answer:- Tautology

v) Negation of p → (p ? ∼ q) is

  1. ∼ p → (∼ p ? q)

  2. p ? (∼ p ? q)

  3. ∼ p ? (∼ p ? ∼ q)

  4. ∼ p → (∼ p → q)

Answer:-

p ? (∼ p ? q)

vi) If p : He is intelligent.
q : He is strong
Then, symbolic form of statement “It is wrong that, he is intelligent or strong” is

  • ∼ p ∨ ∼ p

  • ∼ (p ∧ q)

  • ∼ (p ∨ q)

  • p ∨ ∼ q

Answer:-

∼ (p ∨ q)

vii)A biconditional statement is the conjunction of two ______ statements

  1. Negative
  2. Compound
  3. Connective
  4. Conditional

 

Answer:-

Conditional

ix)  The negation of the statement (p ? q) → (r ? ∼ p)

  1. p ? q ? ∼ r

  2. (p ? q) ? r

  3. p ? q ? ∼ r

  4. (p v q) ? (r ? s)

Answer:-

Contrapositive

ix)  The negation of the statement (p ? q) → (r ? ∼ p)

  1. p ? q ? ∼ r

  2. (p ? q) ? r

  3. p ? q ? ∼ r

  4. (p v q) ? (r ? s)

Answer:-

p ? q ? ∼ r

x)  The false statement in the following is

  1. p ? (∼ p) is contradiction

  2. (p → q) ↔ (∼ q → ∼ p) is a contradiction

  3. ∼ (∼ p) ↔ p is a tautology

  4. p ? (∼ p) ↔ p is a tautology

Answer:-

(p → q) ↔ (∼ q → ∼ p) is a contradiction

Q 2 ) Attempt the following 1 marks

 

i) Find the negation of 10 + 20 = 30

Answer:-

The negation of 10 + 20 = 30 is 10 + 20 ≠ 30

ii) State the truth Value of x2 = 25

Answer:-

x2 = 25’ is an open sentence.

It is not a statement in logic.

iii) Write the negation of p → q

Answer:-

∼p → q ≡ (∼p ∨ q)   .......[? p → q ≡ ∼p ∨ q]

≡ ∼ (∼p) ∧ ∼q       .......[De’Morgan’s Law]

≡ p ∧ ∼q

iv) State the truth value of √3 is not an irrational number

Answer:-

Let p : 3 is irrational number.

∴ Truth value of p is T.

∴ ∼p : 3 is not irrational number.

∴ Truth value of ∼p is F.

v) State the truth value of (p ? ? p)

Answer:-
p ∼ p p ∨ ∼p
T F T
F T T

∴ Truth value of (p ∨ ∼p) is T.

State the truth value of (p ? ∼p)

vi) State the truth value of (p ? ? p)

Answer:-
p ∼p p ∧ ∼p
T F F
F T F

Q3) Attempt the following 2 marks

i) If statements p, q are true and r, s are false, determine the truth values of the following.

~ p ∧ (q ∨ ~ r)

Answer:-

a) Answer :

~ p ∧ (q ∨ ~ r)

≡ ∼T ∧ (T ∨ ∼ F)

≡ F ∧ (T ∨ T)

≡ F ∧ T

≡ F

∴ Truth value of ~ p ∧ (q ∨ ~ r) is F

b) Answer :

(p ∧ ~r) ∧ (~q ∨ s)

≡ (T ∧ ∼F) ∧ (∼T ∨ F)

≡ (T ∧ T) ∧ (F ∨ F)

≡ T ∧ F

≡ F

∴ Truth value of (p ∧ ∼r) ∧ (∼q ∨ s) is F.

ii) Write the following compound statements symbolically.

a) Nagpur is in Maharashtra and Chennai is in Tamilnadu.

b) Triangle is equilateral or isosceles.

Answer:-

a) Answer  :

Let p: Nagpur is in Maharashtra.

Let q: Chennai is in Tamil Nadu.

Then the symbolic form of the given statement is p ∧ q.

b) Answer : 

Let p: Triangle is equilateral.

Let q: Triangle is isosceles.

Then the symbolic form of the given statement is p ∨ q.

iii) . Write the converse and contrapositive of the following statements. “If a function is differentiable then it is continuous”.

Answer:-

Let p: A function is differentiable,

q: It is continuous.

∴ The symbolic form of the given statement is p → q.

Converse: q → p

i.e. If a function is continuous then it is differentiable

Contrapositive: ~q → ~p

i.e. If a function is not continuous then it is not differentiable.

iv) Without using truth table prove that:

~ (p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

Answer:-
I II III IV V VI VII
p q ∼p p ∨ q ∼(p ∨ q) ~(p ∧ q) ~(p ∨ q) ∨ (~p ∧ q)
T T F T F F F
T F F T F F F
F T T T F T T
F F T F T F T

From Columns (III) and (VII), we get ∼(p ∨ q) ∨ (∼p ∧ q) ≡ ∼p

Q4) Answer the following questions

i) Write the negation of the statement “ An angle is a right angle if and only if it is of measure 900 ”

Answer:-

Let p: An angle is a right angle.

q: An angle is of measure 90°.

∴ The symbolic form of the above Statement is p ↔ q.

Note that negation of ‘p ↔ q’ is (p ∧ ∼q) ∨ (q ∧ ∼p).

∴ The negation of the given statement is ‘An angle is a right angle and is not of measure 90° or an angle is of measure 90° and not a right angle.

ii) Write the following statements in symbolic form

a) Milk is white if and only if the sky is not blue

b) If Kutab – Minar is in Delhi then Taj- Mahal is in Agra

c) Even though it is not cloudy , it is still raining

Answer:-

a) Answer :

Let p: Milk is white.

q: Sky is blue.

The given statement in symbolic form is p ↔ ∼q.

b) Answer :

Let p: If Kutub − Minar is in Delhi.

q: Taj − Mahal Is in Agra.

The given statement in symbolic form is p → q.

c) Answer :

Let p: It is cloudy.

q: It is still raining.

∴ The symbolic form of the given statement is ~p ∧ q

iii) Use quantifiers to convert the given open sentence defined on N into a true statement

a) n 2 ≥ 1

b) 3x – 4 < 9

c) Y + 4 > 6

Answer:-

a) Answer :

n2 ≥ 1

∀ n ∈ N, n2 ≥ 1

Since, square of all natural numbers is either 1 or greater than 1.

∴ The statement is true.

b) Answer :

3x – 4 < 9

∃ x ∈ N such that 3x – 4 < 9, is a true statement, since x = 2 ∈ N satisfies 3x – 4 < 9

c) Answer :

Y + 4 > 6

∃ Y ∈ N such that Y + 4 > 6, is a true statement, since Y = 3 ∈ N satisfies Y + 4 > 6.

iv) Examine whether the statement pattern is a tautology, contradiction or contingency

( p ? ? q) → ( ? p ? ? q)

Answer:-
p q ~p ~q p∧~q ~p∧~q (p∧~q)→(~p∧~q)
T T F F F F T
T F F T T F F
F T T F F F T
F F T T F T T

The truth values in the last column are not identical. Hence, it is contingency.

v) Using truth table prove that ? p ? q ≡ ( p ? q ) ? ? p

Answer:-
I II III IV V VI
p q ~p

∼p ? q

p v q

(p v q) ? ∼p

T T F F T F
T F F F T F
F T T T T T
F F T F F F

From column (IV) and (VI), we get

∴ ∼p ? q ≡ (p ? q) ? ∼p

vi) Write the dual of the following

a) 13 is prime number and India is a democratic country

b ) ( p ? ? q ) ? ( ? p ? q ) ≡ ( p ?q ) ? ? ( p ? q)

Answer:-

a) Answer :

13 is prime number or India is a democratic country.

b) Answer :

(p ? ∼q) ? (∼p ? q) ≡ (p ? q) ? ∼(p ? q) 

 

vii) Write the converse, inverse and contrapositive of the statement “If it snows, then they do not drive the car”

 

Answer:-

Let p: It snows.
q: They do not drive the car.

∴ The given statement is p → q.

Its converse is q → p.
If they do not drive the car, then it snows.

Its inverse is ~p → ~q.
If it does not snow, then they drive the car.

Its contrapositive is ~q → ~p.
If they drive the car, then it does not snow.

 

Q5) Answer the following questions

i) Examine whether the statement pattern

[p → (∼q ? r)] ↔ ∼[p → (q → r)] is a tautology, contradiction or contingency.

 

Answer:-

[p → (~q ∨ r)] ↔ ~[p → (q → r)]

p q r ~q ~q ∨ r p →
(~q ∨ r)
q → r p →
(q →r)
~[p →
(q → r)]
[p → (~q ∨ r)]
↔ ~[p → (q → r)]
T T T F T T T T F F
T T F F F F F F T F
T F T T T T T T F F
T F F T T T T T F F
F T T T T T T T F F
F T F F F T F T F F
F F T T T T T T F F
F F F T T T T T F F

All the truth values in the last column are F.

Hence, it is contradiction.

 

ii) Using truth table prove that p ? (q ? r) ≡ (p ? q) ? (p ? r)

Answer:-
I II II IV V VI VII VIII
p q r q ∧ r p ∨ q p ∨ r p ∨ (q ∧ r) (p ∨ q) ∧ (p ∨ r)
T T T T T T T T
T T F F T T T T
T F T F T T T T
T F F F T T T T
F T T T T T T T
F T F F T F F F
F F T F F T F F
F F F F F F F F

From column (VII) and (VIII), we get p ∨ (q ∧ r) ≡ ( p ∨ q) ∧ ( p ∨ r)

iii) Without using truth table show that

(p ? q) ? (∼p v ∼q) ≡ (p v ∼q) ? (∼p v q)

Answer:-

(p ∨ q) ∧ (∼p ? ∼q)

≡ [(p ∨ q) ∧ ∼p] ∨ [(p ∨ q) ∧ ∼q]      .......[Distributive Law]

≡ [(p ∧ ∼p) ∨ (q ∧ ∼p)] ∨ [(p ∧ ∼q) ∨ (q ∧∼q)]  .......[Distributive Law]

≡ [F ∨ (q ∧ ∼p)] ∨ [(p ∧ ∼q) ∨ F]       .......[Complement Law]

≡ (q ∧ ∼p) ∨ (p ∧ ∼q)      .......[Identity Law]

≡ (p ∧ ∼q) ∨ (q ∧ ∼p)     .......[Commutative Law]

iv) With proper justification, state the negation of the following.

 

(p ↔ q) v (~ q → ~r)

Answer:-

~[(p ↔ q) v (~q → ~r)]

≡ ~(p ↔ q) ? (~q → ~r)   ....[Negation of disjunction]

≡ [(p ? ~q) v (q ∧ ~p)] ∧ ~(~q → ~r)  ....[Negation of double implication]

≡ [(p ? ~q) v (q ? ~p)] ? [~ q ? ~(~r)]  ....[Negation of implication]

≡ [(p ? ~q) v (q ? ~p)] ? (~q ? r)   ....[Negation of negation]

v) Prepare truth table for (p ? q) ? ~ r

(p ∧ q) ∨ ~ r

Answer:-

(p ∧ q) ∨ ~ r

p q r ~r p ∧ q (p ∧ q) ∨ ~ r
T T T F T T
T T F T T T
T F T F F F
T F F T F T
F T T F F F
F T F T F T
F F T F F F
F F F T F T