#### Chapters

## Chapter 1: Mathematical Logic

#### Exercise 1.1 [Pages 6 - 8]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.1 [Pages 6 - 8]

State which of the following is the statement. Justify. In case of a statement, state its truth value.

5 + 4 = 13

State which of the following is the statement. Justify. In case of a statement, state its truth value.

x – 3 = 14

State which of the following is the statement. Justify. In case of a statement, state its truth value.

Close the door.

Zero is a complex number.

Please get me breakfast.

Congruent triangles are similar.

x^{2} = x

A quadratic equation cannot have more than two roots.

Do you like Mathematics?

The sunsets in the west

All real numbers are whole numbers.

Can you speak in Marathi?

x^{2} – 6x – 7 = 0, when x = 7

The sum of cube roots of unity is zero.

It rains heavily.

Write the following compound statement symbolically.

Nagpur is in Maharashtra and Chennai is in Tamil Nadu.

Write the following compound statement symbolically.

Triangle is equilateral or isosceles.

Write the following compound statement symbolically.

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

Write the following compound statement symbolically.

Angle is neither acute nor obtuse.

Write the following compound statement symbolically.

If Δ ABC is right-angled at B, then m∠A + m∠C = 90°

Write the following compound statement symbolically.

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

Write the following compound statement symbolically.

x is not irrational number but is a square of an integer.

Write the truth values of the following.

4 is odd or 1 is prime.

Write the truth values of the following.

64 is a perfect square and 46 is a prime number.

Write the truth values of the following.

5 is a prime number and 7 divides 94.

Write the truth values of the following.

It is not true that 5−3i is a real number.

Write the truth value of the following.

If 3 × 5 = 8 then 3 + 5 = 15.

Write the truth value of the following.

Milk is white if and only if sky is blue.

Write the truth values of the following.

24 is a composite number or 17 is a prime number.

If the statement p, q are true statement and r, s are false statement then determine the truth value of the following:

p ∨ (q ∧ r)

If the statement p, q are true statement and r, s are false statement then determine the truth value of the following:

(p → q) ∨ (r → s)

If the statement p, q are true statement and r, s are false statement then determine the truth value of the following:

(q ∧ r) ∨ (∼ p ∧ s)

(p → q) ∧ ∼ r

(∼ r ↔ p) → ∼ q

[∼ p ∧ (∼ q ∧ r)] ∨ [(q ∧ r) ∨ (p ∧ r)]

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

∼ [(∼ p ∧ r) ∨ (s → ∼ q)] ↔ (p ∧ r)

Write the negation of the following.

Tirupati is in Andhra Pradesh.

Write the negation of the following.

3 is not a root of the equation x^{2} + 3x – 18 = 0

Write the negation of the following.

`sqrt2` is a rational number.

Write the negation of the following.

Polygon ABCDE is a pentagon.

Write the negation of the following.

#### Exercise 1.2 [Page 13]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.2 [Page 13]

Construct the truth table of the following statement pattern.

[(p → q) ∧ q] → p

Construct the truth table of the following statement pattern.

(p ∧ ∼q) ↔ (p → q)

Construct the truth table of the following statement pattern.

(p ∧ q) ↔ (q ∨ r)

Construct the truth table of the following statement pattern.

p → [∼ (q ∧ r)]

Construct the truth table of the following statement pattern.

∼ p ∧ [(p ∨ ∼ q) ∧ q]

Construct the truth table of the following statement pattern.

(∼ p → ∼ q) ∧ (∼ q → ∼ p)

Construct the truth table of the following statement pattern.

(q → p) ∨ (∼ p ↔ q)

Construct the truth table of the following statement pattern.

[p → (q → r)] ↔ [(p ∧ q) → r]

Construct the truth table of the following statement pattern.

p → [∼ (q ∧ r)]

Construct the truth table of the following statement pattern.

(p ∨ ∼ q) → (r ∧ p)

Using truth table, prove that ~ p ∧ q ≡ (p ∨ q) ∧ ~ p

Using the truth table prove the following logical equivalence.

∼ (p ∨ q) ∨ (∼ p ∧ q) ≡ ∼ p

Using the truth table prove the following logical equivalence.

p ↔ q ≡ ∼ [(p ∨ q) ∧ ∼ (p ∧ q)]

Using the truth table prove the following logical equivalence.

p → (q → p) ≡ ∼ p → (p → q)

Using the truth table prove the following logical equivalence.

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

Using the truth table prove the following logical equivalence.

p → (q ∧ r) ≡ (p → q) ∧ (p → r)

Using the truth table prove the following logical equivalence.

p → (q ∧ r) ≡ (p ∧ q) (p → r)

Using the truth table prove the following logical equivalence.

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Using the truth table prove the following logical equivalence.

[∼ (p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

Using the truth table proves the following logical equivalence.

∼ (p ↔ q) ≡ (p ∧ ∼ q) ∨ (q ∧ ∼ p)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p ∧ q) → (q ∨ p)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p → q) ↔ (∼ p ∨ q)

Discuss the statement pattern, using truth table : ~(~p ∧ ~q) v q

Examine whether the following logical statement pattern is a tautology, contradiction, or contingency.

[(p → q) ∧ q] → p

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

[(p → q) ∧ ∼ q] → ∼ p

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p ↔ q) ∧ (p → ∼ q)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

∼ (∼ q ∧ p) ∧ q

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p ∧ ∼ q) ↔ (p → q)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(∼ p → q) ∧ (p ∧ r)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

[p → (∼ q ∨ r)] ↔ ∼ [p → (q → r)]

#### Exercise 1.3 [Pages 17 - 18]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.3 [Pages 17 - 18]

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of the following.

∃ x ∈ A such that x – 8 = 1

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of the following.

∀ x ∈ A, x^{2} + x is an even number

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of the following.

∃ x ∈ A such that x^{2} < 0

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of the following.

∀ x ∈ A, x is an even number

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of the following.

∃ x ∈ A such that 3x + 8 > 40

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of the following.

∀ x ∈ A, 2x + 9 > 14

Write the dual of the following.

p ∨ (q ∧ r)

Write the dual of the following.

p ∧ (q ∧ r)

Write the dual of the following.

(p ∨ q) ∧ (r ∨ s)

Write the dual of the following.

p ∧ ∼ q

Write the dual of the following.

(∼ p ∨ q) ∧ (∼ r ∧ s)

Write the dual of the following.

∼ p ∧ (∼ q ∧ (p ∨ q) ∧ ∼ r)

Write the dual of the following.

[∼ (p ∨ q)] ∧ [p ∨ ∼ (q ∧ ∼ s)]

Write the dual of the following.

c ∨ {p ∧ (q ∨ r)}

Write the dual of the following.

∼ p ∨ (q ∧ r) ∧ t

Write the dual of the following.

(p ∨ q) ∨ c

Write the negation of the following.

x + 8 > 11 or y – 3 = 6

Write the negation of the following.

11 < 15 and 25 > 20

Write the negation of the following.

Quadrilateral is a square if and only if it is a rhombus.

Write the negation of the following.

It is cold and raining.

Write the negation of the following.

If it is raining then we will go and play football.

Write the negation of the following.

`sqrt2` is a rational number.

Write the negation of the following.

All-natural numbers are whole numbers.

Write the negation of the following.

∀ n ∈ N, n^{2 }+ n + 2 is divisible by 4.

Write the negation of the following.

∃ x ∈ N such that x – 17 < 20

Write converse, inverse and contrapositive of the following statement.

If x < y then x^{2} < y^{2} (x, y ∈ R)

Write converse, inverse and contrapositive of the following statement.

A family becomes literate if the woman in it is literate.

Write converse, inverse and contrapositive of the following statement.

If surface area decreases then pressure increases.

Write converse, inverse and contrapositive of the following statement.

If voltage increases then current decreases.

#### Exercise 1.4 [Page 21]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.4 [Page 21]

Using the rule of negation write the negation of the following with justification.

∼ q → p

Using the rule of negation write the negation of the following with justification.

p ∧ ∼ q

Using the rule of negation write the negation of the following with justification.

p ∨ ∼ q

Using the rule of negation write the negation of the following with justification.

(p ∨ ∼ q) ∧ r

Using the rule of negation write the negation of the following with justification.

p → (p ∨ ∼ q)

Using the rule of negation write the negation of the following with justification.

∼ (p ∧ q) ∨ (p ∨ ∼ q)

Using the rule of negation write the negation of the following with justification.

(p ∨ ∼ q) → (p ∧ ∼ q)

Using the rule of negation write the negation of the following with justification.

(∼ p ∨ ∼ q) ∨ (p ∧ ∼ q)

Rewrite the following statement without using if ...... then.

If a man is a judge then he is honest.

Rewrite the following statement without using if ...... then.

It 2 is a rational number then `sqrt2` is irrational number.

Rewrite the following statement without using if ...... then.

It f(2) = 0 then f(x) is divisible by (x – 2).

Without using the truth table show that P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)

Without using truth table prove that:

(p ∨ q) ∧ (p ∨ ∼ q) ≡ p

Without using truth table prove that:

(p ∧ q) ∨ (∼ p ∧ q) ∨ (p ∧ ∼ q) ≡ p ∨ q

Without using truth table prove that:

∼ [(p ∨ ∼ q) → (p ∧ ∼ q)] ≡ (p ∨ ∼ q) ∧ (∼ p ∨ q)

#### Exercise 1.5 [Pages 29 - 30]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.5 [Pages 29 - 30]

Express the following circuit in the symbolic form of logic and writ the input-output table.

Express the following circuit in the symbolic form of logic and writ the input-output table.

Express the following circuit in the symbolic form of logic and writ the input-output table.

Express the following circuit in the symbolic form of logic and writ the input-output table.

Express the following circuit in the symbolic form of logic and writ the input-output table.

Express the following circuit in the symbolic form of logic and writ the input-output table.

Construct the switching circuit of the following:

(∼ p ∧ q) ∨ (p ∧ ∼ r)

Construct the switching circuit of the following:

(p ∧ q) ∨ [∼ p ∧ (∼ q ∨ p ∨ r)]

Construct the switching circuit of the following:

(p ∧ r) ∨ (∼ q ∧ ∼ r)] ∧ (∼ p ∧ ∼ r)

Construct the switching circuit of the following:

(p ∧ ∼ q ∧ r) ∨ [p ∧ (∼ q ∨ ∼ r)]

Construct the switching circuit of the following:

p ∨ (∼ p) ∨ (∼ q) ∨ (p ∧ q)

Construct the switching circuit of the following:

(p ∧ q) ∨ (∼ p) ∨ (p ∧ ∼ q)

Give an alternative equivalent simple circuit for the following circuit:

Give an alternative equivalent simple circuit for the following circuit:

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

Write the symbolic form of the following switching circuit construct its switching table and interpret it.

Write the symbolic form of the following switching circuit construct its switching table and interpret it.

Obtain the simple logical expression of the following. Draw the corresponding switching circuit.

p ∨ (q ∧ ∼ q)

Obtain the simple logical expression of the following. Draw the corresponding switching circuit.

(∼ p ∧ q) ∨ (∼ p ∧ ∼ q) ∨ (p ∧ ∼ q)

Obtain the simple logical expression of the following. Draw the corresponding switching circuit.

[p ∨ ( ∼ q) ∨ (∼ r)] ∧ [p ∨ (q ∧ r)]

Obtain the simple logical expression of the following. Draw the corresponding switching circuit.

(p ∧ q ∧ ∼ p) ∨ (∼ p ∧ q ∧ r) ∨ (p ∧ q ∧ r)

#### Miscellaneous Exercise 1 [Pages 32 - 35]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Miscellaneous Exercise 1 [Pages 32 - 35]

Select and write the correct answer from the given alternative of the following question:

If p ∧ q is false and p ∨ q is true, then __________ is not true.

p ∨ q

p ↔ q

∼p ∨ ∼q

q ∨ ∼p

Select and write the correct answer from the given alternative of the following question:

(p ∧ q) → r is logically equivalent to ________ .

p → (q → r)

(p ∧ q) → ∼r

(∼p ∨ ∼q) → ∼r

(p ∨ q) → r

Select and write the correct answer from the given alternative of the following question:

Inverse of statement pattern (p ∨ q) → (p ∧ q) is ________ .

(p ∧ q) → (p ∨ q)

∼ (p ∨ q) → (p ∧ q)

(∼p ∧ ∼q) → (∼p ∨ ∼q)

(∼p ∨ ∼q) → (∼p ∧ ∼q)

Select and write the correct answer from the given alternative of the following question:

If p ∧ q is F, p → q is F then the truth values of p and q are ________.

T, T

T, F

F, T

F, F

Select and write the correct answer from the given alternative of the following question:

The negation of inverse of ∼p → q is ________.

q ∧ p

∼p ∧ ∼q

p ∧ q

∼q → ∼p

Select and write the correct answer from the given alternative of the following question:

The negation of p ∧ (q → r) is ________.

∼p ∧ (∼q → ∼r)

p ∨ (∼q ∨ r)

∼p ∧ (∼q → ∼r)

∼p ∨ (∼q ∧ ∼r)

Select and write the correct answer from the given alternative of the following question:

If A = {1, 2, 3, 4, 5} then which of the following is not true?

∃ x ∈ A such that x + 3 = 8

∃ x ∈ A such that x + 2 < 9

∀ x ∈ A, x + 6 ≥ 9

∃ x ∈ A such that x + 6 < 10

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

4! = 24.

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

π is an irrational number.

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

India is a country and Himalayas is a river.

Please get me a glass of water.

cos^{2}θ − sin^{2}θ = cos2θ for all θ∈R.

If x is a whole number then x + 6 = 0.

Write the truth value of the following statement:

`sqrt5` is an irrational but `3sqrt5` is a complex number.

Write the truth value of the following statement:

∀ n ∈ N, n^{2} + n is even number while n^{2} – n is an odd number.

Write the truth value of the following statement:

∃ n ∈ N such that n + 5 > 10.

Write the truth value of the following statement:

The square of any even number is odd or the cube of any odd number is odd.

Write the truth value of the following statement:

In ΔABC if all sides are equal then its all angles are equal.

Write the truth value of the following statement:

∀ n ∈ N, n + 6 > 8.

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∃ x ∈ A such that x + 8 = 15

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∀ x ∈ A, x + 5 < 12.

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∃ x ∈ A, such that x + 7 ≥ 11.

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, determine the truth value of the following statement:

∀ x ∈ A, 3x ≤ 25.

Write the negation of the following:

∀ n ∈ A, n + 7 > 6.

Write the negation of the following:

∃ x ∈ A, such that x + 9 ≤ 15.

Write the negation of the following:

Some triangles are equilateral triangle.

Construct the truth table of the following:

p → (q → p)

Construct the truth table of the following:

(∼p ∨ ∼q) ↔ [∼(p ∧ q)]

Construct the truth table of the following:

∼ (∼p ∧ ∼q) ∨ q

Construct the truth table of the following:

[(p ∧ q) ∨ r] ∧ [∼r ∨ (p ∧ q)]

Construct the truth table of the following:

[(∼p ∨ q) ∧ (q → r)] → (p → r)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

[(p → q) ∧ ∼ q] → ∼ p

Determine whether the following statement pattern is a tautology, contradiction, or contingency:

[(p ∨ q) ∧ ∼p] ∧ ∼q

Determine whether the following statement pattern is a tautology, contradiction, or contingency:

(p → q) ∧ (p ∧ ∼q)

Determine whether the following statement pattern is a tautology, contradiction or contingency:

[p → (q → r)] ↔ [(p ∧ q) → r]

Determine whether the following statement pattern is a tautology, contradiction or contingency:

[(p ∧ (p → q)] → q

Determine whether the following statement pattern is a tautology, contradiction or contingency:

(p ∧ q) ∨ (∼p ∧ q) ∨ (p ∨ ∼q) ∨ (∼p ∧ ∼q)

Determine whether the following statement pattern is a tautology, contradiction or contingency:

[(p ∨ ∼q) ∨ (∼p ∧ q)] ∧ r

Determine whether the following statement pattern is a tautology, contradiction or contingency:

(p → q) ∨ (q → p)

Determine the truth values of p and q in the following case:

(p ∨ q) is T and (p ∧ q) is T

Determine the truth values of p and q in the following case:

(p ∨ q) is T and (p ∨ q) → q is F

Determine the truth values of p and q in the following case:

(p ∧ q) is F and (p ∧ q) → q is T

Using the truth table, prove the following logical equivalence :

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)

Using truth table, prove the following logical equivalence :

(p ∧ q) → r ≡ p → (q → r)

Using rules in logic, prove the following:

p ↔ q ≡ ∼(p ∧ ∼q) ∨ ∼(q ∧ ∼p)

Using rules in logic, prove the following:

∼p ∧ q ≡ (p ∨ q) ∧ ∼p

Using rules in logic, prove the following:

∼ (p ∨ q) ∨ (∼p ∧ q) ≡ ∼p

Using the rules in logic, write the negation of the following:

(p ∨ q) ∧ (q ∨ ∼r)

Using the rules in logic, write the negation of the following:

p ∧ (q ∨ r)

Using the rules in logic, write the negation of the following:

(p → q) ∧ r

Using the rules in logic, write the negation of the following:

(∼p ∧ q) ∨ (p ∧ ∼q)

Express the following circuit in the symbolic form. Prepare the switching table:

Express the following circuit in the symbolic form. Prepare the switching table:

Simplify the following so that the new circuit has a minimum number of switches. Also, draw the simplified circuit.

Simplify the following so that the new circuit has a minimum number of switches. Also, draw the simplified circuit.

Check whether the following switching circuits are logically equivalent - Justify.

(i)

(ii)

Check whether the following switching circuits are logically equivalent - Justify.

(i)

(ii)

Give alternative arrangement of the switching following circuit, has minimum switches.

Simplify the following so that the new circuit.

Represent the following switching circuit in symbolic form and construct its switching table. Write your conclusion from the switching table.

## Chapter 1: Mathematical Logic

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 1 - Mathematical Logic

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 1 (Mathematical Logic) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 1 Mathematical Logic are Truth Value of Statement in Logic, Logical Connective, Simple and Compound Statements, Truth Tables of Compound Statements, Statement Patterns and Logical Equivalence, Tautology, Contradiction, and Contingency, Quantifier and Quantified Statements in Logic, Duality, Negations of Compound Statements, Converse, Inverse, and Contrapositive, Algebra of Statements, Application of Logic to Switching Circuits, Switching Table..

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