# 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

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

a) Contingency

c) Tautology

d) None of these

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

b) Tautology

c) Neither (i) nor (ii)

d) None of these

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

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

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

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

## 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)

## 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 Fb) 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 → pi.e. If a function is continuous then it is differentiableContrapositive: ~q → ~pi.e. If a function is not continuous then it is not differentiable.

iv) Without using truth table prove that:

~ (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 ≥ 1Since, 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 < 9c) 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.

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