Mathematical Logic - Ex 1.1 | Maharashtra Board 12th Maths Solutions Chapter 1

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 1. - State which of the following sentences are statements. Justify your answer. In case of statement, write down the truth value :

(i) 5 + 4 = 13.

Solution:

It is a statement which is false, hence its truth value is ‘F’.

(ii) x – 3 = 14.

Solution:

It is an open sentence, hence it is not a statement.

(iii) Close the door.

Solution:

It is an imperative sentence, hence it is not a statement.

(iv) Zero is a complex number.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(v) Please get me breakfast.

Solution:

It is an imperative sentence, hence it is not a statement.

(vi) Congruent triangles are also similar.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(vii) x2 = x.

Solution:

It is an open sentence, hence it is not a statement,

(viii) A quadratic equation cannot have more than two roots.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(ix) Do you like Mathematics ?

Solution:

It is an interrogative sentence, hence it is not a statement.

(x) The sun sets in the west.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(xi) All real numbers are whole numbers.

Solution:

It is a statement which is false, hence its truth value is ‘F’.

(xii) Can you speak in Marathi ?

Solution:

It is an interrogative sentence, hence it is not a statement.

(xiii) x2 – 6x – 7 = 0, when x = 7.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(xiv) The sum of cuberoots of unity is zero.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(xv) It rains heavily.

Solution :

It is an open sentence, hence it is not a statement.

Question 2. Write the following compound statements symbolically:

(i) Nagpur is in Maharashtra and Chennai is in Tamil Nadu.

Solution:

Let p : Nagpur is in Maharashtra.

q : Chennai is in Tamil Nadu.

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

(ii) Triangle is equilateral or isosceles,

Solution:

Let p : Triangle is equilateral.

q : Triangle is isosceles.

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

(iii) The angle is right angle if and only if it is of measure 90°.

Solution:

Let p : The angle is right angle.

q : It is of measure 90°.

Then the symbolic form of the given statement is p↔q

(iv) Angle is neither acute nor obtuse.

Solution:

Let p : Angle is acute.

q : Angle is obtuse.

Then the symbolic form of the given statement is

~p ∧ ~q.

(v) If ∆ ABC is right angled at B, then m∠A + m∠C = 90°.

Solution:

Let p : ∆ ABC is right angled at B.

q : m∠A + m∠C = 90°.

Then the symbolic form of the given statement is p → q

(vi) Hima Das wins gold medal if and only if she runs fast.

Solution:

Let p : Hima Das wins gold medal

q : She runs fast.

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

(vii) x is not irrational number but it is a square of an integer.

Solution:

Let p : x is not irrational number

q : It is a square of an integer

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

Note : If p : x is irrational number, then the symbolic form of the given statement is ~p ∧ q.

Question 3. Write the truth values of the following :

(i) 4 is odd or 1 is prime.

Solution:

Let p : 4 is odd.

q : 1 is prime.

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

The truth values of both p and q are F.

∴ the truth value of p v q is F. … [F ∨ F = F]

(ii) 64 is a perfect square and 46 is a prime number.

Solution:

Let p : 64 is a perfect square.

q : 46 is a prime number.

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

The truth values of p and q are T and F respectively.

∴ the truth value of p ∧ q is F. … [T ∧ F ≡ F]

(iii) 5 is a prime number and 7 divides 94.

Solution:

Let p : 5 is a prime number.

q : 7 divides 94.

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

The truth values of p and q are T and F respectively.

∴ the truth value of p ∧ q is F. … [T ∧ F ≡ F]

(iv) It is not true that 5 – 3i is a real number.

Solution:

Let p : 5 – 3i is a real number.

Then the symbolic form of the given statement is ~ p.

The truth values of p is F.

∴ the truth values of ~ p is T. … [~ F ≡ T]

(v) If 3 × 5 = 8, then 3 + 5 = 15.

Solution:

Let p : 3 × 5 = 8.

q : 3 + 5 = 15.

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

The truth values of both p and q are F.

∴ the truth value of p → q is T. … [F → F ≡ T]

(vi) Milk is white if and only if sky is blue.

Solution:

Let p : Milk is white.

q : Sky is blue

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

The truth values of both p and q are T.

∴ the truth value of p ↔ q is T. … [T ↔ T ≡ T]

(vii) 24 is a composite number or 17 is a prime number.

Solution :

Let p : 24 is a composite number.

q : 17 is a prime number.

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

The truth values of both p and q are T.

∴ the truth value of p ∨ q is T. … [T ∨ T ≡ T]

Question 4.If the statements p, q are true statements and r, s are false statements, then determine the truth values of the following:

(i) p ∨ (q ∧ r)

Solution:

Truth values of p and q are T and truth values of r and s are F.

p ∨ (q ∧ r) ≡ T ∨ (T ∧ F)

≡ T ∧ F ≡ T

Hence the truth value of the given statement is true.

(ii) (p → q) ∨ (r → s)

Solution:

(p → q) ∨ (r → s) ≡ (T → T) ∨ (F → F)

≡ T ∨ T ≡ T

Hence the truth value of the given statement is true.

(iii) (q ∧ r) ∨ (~p ∧ s)

Solution:

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

≡ F ∨ (F ∧ F)

≡ F ∨ F ≡ F

Hence the truth value of the given statement is false.

(iv) (p → q) ∧ (~ r)

Solution:

(p → q) ∧ (~ r) ≡ (T → T) ∧ (~ F)

≡ T ∧ T ≡ T

Hence the truth value of the given statement is true.

(v) (~r ↔ p) → (~q)

Solution:

(~r ↔ p) → (~q) ≡ (~F ↔ T) → (~T)

≡ (T ↔ T) → F

≡ T → F ≡ F

Hence the truth value of the given statement is false.

(vi) [~p ∧ (~q ∧ r) ∨ (q ∧ r) ∨ (p ∧ r)]

Solution:

[~p ∧ (~q ∧ r)∨(q ∧ r)∨(p ∧ r)]

≡ [~T ∧ (~T ∧ F)] ∨ [(T ∧ F) V (T ∧ F)]

≡ [F ∧ (F ∧ F)] ∨ [F V F]

≡ (F ∧ F) ∨ F

≡ F ∨ F ≡ F

Hence the truth value of the given statement is false.

(vii) [(~ p ∧ q) ∧ (~ r)] ∨ [(q → p) → (~ s ∨ r)]

Solution:

[(~ p ∧ q) ∧ (~ r)] ∨ [(q → p) → (~ s ∨ r)]

≡ [(~T ∧ T) ∧ (~F)] ∨ [(T → T) → (~F ∨ F)]

≡ [(F ∧ T) ∧ T] ∨ [T → (T ∨ F)]

≡ (F ∧ T) ∨ (T → T)

≡ F ∨ T ≡ T

Hence the truth value of the given statement is true.

(viii) ~ [(~p ∧ r) ∨ (s → ~q)] ↔ (p ∧ r)

Solution :

~ [(~p ∧ r) ∨ (s → ~q)] ↔ (p ∧ r)

≡ ~ [(~T ∧ F) ∨ (F → ~T)] ↔ (T ∧ F)

≡ ~ [(F ∧ F) ∨ (F → F)] ↔ F

≡ ~ (F ∨ T) ↔ F

≡ ~T ↔ F

≡ F ↔ F ≡ T

Hence the truth value of the given statement is true.

Question 5. Write the negations of the following :

(i) Tirupati is in Andhra Pradesh.

Solution:

The negations of the given statements are :

Tirupati is not in Andhra Pradesh.

(ii) 3 is not a root of the equation x2 + 3x – 18 = 0.

Solution:

3 is a root of the equation x2 + 3x – 18 = 0.

(iii) 2–√ is a rational number.

Solution:

2–√ is not a rational number.

(iv) Polygon ABCDE is a pentagon.

Solution:

Polygon ABCDE is not a pentagon.

(v) 7 + 3 > 5.

Solution :

7 + 3 > 5.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

1.1.1 Introduction :

Mathematics is a logical subject and tries to be exact. For exactness, it requires proofs which depend upon proper reasoning. Reasoning requires logic. The word Logic is derived from the Greek word "LOGOS" which means reason. Therefore logic deals with the method of reasoning.

In ancient Greece the great philosopher and thinker Aristotle started study of Logic systematically. In mathematics Logic has been developed by English Philosopher and mathematician George Boole (2 November 1815 - 8 December 1864) Language is the medium of communication of our thoughts. For communication we use sentences. In logic, we use the statements which are special sentences.

1.1.2 Statement : A statement is a declarative (assertive) sentence which is either true or false, but not both simultaneously. Statements are denoted by p, q, r, .....

1.1.3 Truth value of a statement : Each statement is either true or false. If a statement is true then its truth value is 'T' and if the statement is false then its truth value is F.

Illustrations :

1) Following sentences are statements.

i) Sun rises in the East.

ii) 5 × 2 = 11

iii) Every triangle has three sides

iv) Mumbai is the capital of Maharashtra.

v) Every equilateral triangle is an equiangular triangle.

vi) A natural number is an integer.

2) Following sentences are not statements.

i) Please, give your Pen.

ii) What is your name ?

iii) What a beautiful place it is !

iv) How are you ?

v) Do you like to play tennis ?

vi) Open the window.

vii) Let us go for tea

viii) Sit down.

Note : Interrogative, exclamatory, command, order, request, suggestion are not statements.

3) Consider the following.

i) 32

9 0 x − =

ii) He is tall.

iii) Mathematics is an interesting subject.

iv) It is black in colour.

Let us analyse these statements.

i) For x = 6 it is true but for other than 6 it is not true.

ii) Here, we cannot determine the truth value.

For

iii) & iv) the truth value varies from person to person. In all the above sentences, the truth value depends upon the situation. Such sentences are called as open sentences. Open sentence is not a statement.

Q.1. Which of the following sentences are statements in logic ? Write down the truth values of

the statements.

i) 6 × 4 = 25

ii) x + 6 = 9

iii) What are you doing ?

iv) The quadratic equation x2

– 5x + 6 = 0 has 2 real roots.

v) Please, sit down

vi) The Moon revolves around the earth.

vii) Every real number is a complex number.

viii) He is honest.

ix) The square of a prime number is a prime number.

Solution :

i) It is a statement which is false, hence its truth value is F.

ii) It is an open sentence hence it is not a statement.

iii) It is an interrogative hence it is not a statement.

iv) It is a statement which is true hence its truth value is T.

v) It is a request hence it is not a statement.

vi) It is a statement which is true, hence its truth value is T.

vii) It is a statement which is true, hence its truth value is T.

viii) It is open sentence, hence it is not a statement.

ix) It is a statement which is false, hence its truth value is F.

1.1.4 Logical connectives, simple and compound statements :

The words or phrases which are used to connect two statements are called logical connectives. We will study the connectives 'and', 'or', 'if ..... then', 'if and only if ', 'not''. Simple and Compound Statements : A statement which cannot be split further into two or more statements is called a simple statement. If a statement is the combination of two or more simple statements, then it is called a compound statement.

"3 is a prime and 4 is an even number", is a compound statement.
"3 and 5 are twin primes", is a simple statement.

We describe some connectives

1) Conjunction : If two statements are combined using the connective 'and' then it is called as a conjunction. In other words if p, q are two statements then 'p and q' is called as conjunction. It is denoted by 'p ∧ q' and it is read as 'p conjunction q' or 'p and q'. The conjunction p ∧ q is said to be true if and only if both p and q are true.

2) Disjunction : If two statements are combined by using the logical connective 'or' then it is called as a disjunction. In other words if p, q are two staements then 'p or q' is called as disjunction. It is denoted by 'p ∨ q and it is read as 'p or q' or 'p disjunction q'.

3) Conditional (Implication) : If two statements are combined by using the connective. 'if .... then', then it is called as conditional or implication. In other words if p, q are two statements then 'if p then q' is called as conditional. It is denoted by p → q or p ⇒ q and it is read as 'p implies q' or 'if p then q'.

4) Biconditional (Double implication) :

If two statements are combined using the logical connective 'if and only if ' then it is called as biconditional. In other words if p, q are two statements then 'p if and only if q' is called as biconditional. It is denoted by 'p ↔ q' or p ⇔ q. It is read as 'p biconditional q' or 'p if and only if q'.

5) Negation of a statement : For any given statement p, there is another statement which is defined to be true when p is false, and false when p is true, is called the negation of p and is denoted by ~p.

Ex.1:Express the following compound statements symbolically without examining the truth values.

i) 2 is an even number and 25 is a perfect square.

ii) A school is open or there is a holiday.

iii) Delhi is in India but Dhaka is not in Srilanks.

iv) 3 + 8 ≥ 12 if and only if 5 × 4 ≤ 25.

Solution :

i) Let p : 2 is an even numder

q : 25 is a perfect square.

The symbolic form is p ∧ q.

ii) Let p : The school is open

q : There is a holiday

The symbolic form is p ∨ q

iii) Let p : Delhi is in India

q : Dhaka is in Srilanka

The symbolic form is p ∧ ~ q.

iv) Let p : 3 + 8 ≥ 12; q : 5 × 4 ≤ 25

The symbolic form is p ↔ q

Ex.2.Write the truth values of the following statements.

i) 3 is a prime number and 4 is a rational number.

ii) All flowers are red or all cows are black.

iii) If Mumbai is in Maharashtra then Delhi is the capital of India.

iv) Milk is white if and only if the Sun rises in the West.

Solution :

i) Let p : 3 is a prime number

q : 4 is a rational number.

Truth values of p and q are T and T respectively.

The given statement in symbolic form is p ∧ q.

The truth value of given statement is T.

ii) Let p : All flowers are red ; q : All cows are black.

Truth values of p and q are F and F respectively.

The given statement in the symbolic form is p ∨ q

\ p ∨ q ≡ F ∨ F is F

\ Truth value of given statement is F.

iii) Let p : Mumbai is in Maharashtra

q : Delhi is capital of India

Truth values of p and q are T and T respectively.

The given statement in symbolic form is p → q

\ p → q ≡ T → T is T

\Truth value of given statement is T

iv) Let p : Milk is white; q : Sun rises in the West.

Truth values of p and q are T and F respectively.

The given statement in symbolic form is p ↔ q

\ p ↔ q ≡ T ↔ F is F

\Truth value of given statement is F

Ex.3 : If statements p, q are true and r, s are false, determine the truth values of the following.

i) ~ p ∧ (q ∨ ~ r) ii) (p ∧ ~ r) ∧ (~ q ∨ s)

iii) ~ (p → q) ↔ (r ∧ s) iv) (~p → q) ∧ (r ↔ s)

Solution :

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

Hence truth value is F.

ii) (p ∧ ~ r) ∧ (~ q ∨ s) ≡ (T ∧ ~ F) ∧ (~T ∨ F) ≡ (T ∧ T) ∧ (F ∨ F) ≡ T ∧ F ≡ F.

Hence truth value is F.

iii) [~(p → q)] ↔ (r ∧ s) ≡ [~ ( T → T )] ↔ (F ∧ F) ≡ (~ T) ↔ (F) ≡ F ↔ F ≡ T.

Hence truth value is T

iv) (~p → q) ∧ (r ↔ s) ≡ (~T → T) ∧ (F ↔ F) ≡ (F →T) ∧ T ≡ T ∧ T ≡ T.

Hence truth value is T .

Ex.4.Write the negations of the following.

i) Price increases

ii) 0! ≠ 1

iii) 5 + 4 = 9

Solution :

i) Price does not increase

ii) 0! = 1

iii) 5 + 4 ≠ 9