Tamil Nadu Board of Secondary EducationHSC Science Class 12th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 12 - Discrete Mathematics [Latest edition]

## Chapter 12: Discrete Mathematics

Exercise 12.1Exercise 12.2Exercise 12.3
Exercise 12.1 [Pages 235 - 236]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 12 Discrete MathematicsExercise 12.1 [Pages 235 - 236]

Exercise 12.1 | Q 1. (i) | Page 235

Determine whether * is a binary operation on the sets-given below.

a * b – a.|b| on R

Exercise 12.1 | Q 1. (ii) | Page 235

Determine whether * is a binary operation on the sets-given below.

a * b = min (a, b) on A = {1, 2, 3, 4, 5}

Exercise 12.1 | Q 1. (iii) | Page 235

Determine whether * is a binary operation on the sets-given below.

(a * b) = "a"sqrt("b") is binary on R

Exercise 12.1 | Q 2 | Page 235

On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?

Exercise 12.1 | Q 3 | Page 235

Let * be defined on R by (a * b) = a + b + ab – 7. Is * binary on R? If so, find 3 * ((-7)/15)

Exercise 12.1 | Q 4 | Page 235

Let A = {a + sqrt(5)b : a, b ∈ Z}. Check whether the usual multiplication is a binary operation on A

Exercise 12.1 | Q 5. (i) | Page 235

Define an operation * on Q as follows: a * b = (("a" + "b")/2); a, b ∈ Q. Examine the closure, commutative and associate properties satisfied by * on Q.

Exercise 12.1 | Q 5. (ii) | Page 235

Define an operation * on Q as follows: a * b = (("a" + "b")/2); a, b ∈ Q. Examine the existence of identity and the existence of inverse for the operation * on Q.

Exercise 12.1 | Q 6 | Page 236

Fill in the following table so that the binary operation * on A = {a, b, c} is commutative.

 * a b c a b b c b a c a c
Exercise 12.1 | Q 7 | Page 236

Consider the binary operation * defined on the set A = {a, b, c, d} by the following table:

 * a b c d a a c b d b d a b c c c d a a d d b a c

Is it commutative and associative?

Exercise 12.1 | Q 8. (i) | Page 236

Let A = ((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1)), B = ((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1)), C = ((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1)) be any three boolean matrices of the same type. Find A v B

Exercise 12.1 | Q 8. (ii) | Page 236

Let A = ((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1)), B = ((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1)), C = ((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1)) be any three boolean matrices of the same type. Find A ∧ B

Exercise 12.1 | Q 8. (iii) | Page 236

Let A = ((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1)), B = ((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1)), C = ((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1)) be any three boolean matrices of the same type. Find (A v B) ∧ C

Exercise 12.1 | Q 8. (iv) | Page 236

Let A = ((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1)), B = ((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1)), C = ((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1)) be any three boolean matrices of the same type. Find (A ∧ B) v C

Exercise 12.1 | Q 9. (i) | Page 236

Let M = {{:((x, x),(x, x)) : x ∈ "R"- {0}:}} and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the commutative and associative properties satisfied by * on M

Exercise 12.1 | Q 9. (ii) | Page 236

Let M = {{:((x, x),(x, x)) : x ∈ "R"- {0}:}} and let * be the matrix multiplication. Determine whether M is closed under * . If so, examine the existence of identity, existence of inverse properties for the operation * on M

Exercise 12.1 | Q 10. (i) | Page 236

Let A be Q\{1} Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the commutative and associative properties satisfied by * on A

Exercise 12.1 | Q 10. (ii) | Page 236

Let A be Q\{1}. Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the existence of an identity, the existence of inverse properties for the operation * on A

Exercise 12.2 [Pages 248 - 249]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 12 Discrete MathematicsExercise 12.2 [Pages 248 - 249]

Exercise 12.2 | Q 1. (i) | Page 248

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

¬ P

Exercise 12.2 | Q 1. (ii) | Page 248

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

P ∧ ¬q

Exercise 12.2 | Q 1. (iii) | Page 248

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

¬p v q

Exercise 12.2 | Q 1. (iv) | Page 248

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

p → ¬q

Exercise 12.2 | Q 1. (v) | Page 248

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

p ↔ q

Exercise 12.2 | Q 2. (i) | Page 248

Write the following sentences in symbolic form using statement variables p and q.

19 is not a prime number and all the angles of a triangle are equal

Exercise 12.2 | Q 2. (ii) | Page 248

Write the following sentences in symbolic form using statement variables p and q.

19 is a prime number or all the angles of a triangle are not equal

Exercise 12.2 | Q 2. (iii) | Page 248

Write the following sentences in symbolic form using statement variables p and q.

19 is a prime number and all the angles of a triangle are equal

Exercise 12.2 | Q 2. (iv) | Page 248

Write the following sentences in symbolic form using statement variables p and q.

19 is not a prime number

Exercise 12.2 | Q 3. (i) | Page 248

Determine the truth value of the following statement.

If 6 + 2 = 5, then the milk is white.

Exercise 12.2 | Q 3. (ii) | Page 248

Determine the truth value of the following statement.

China is in Europe dr sqrt(3) is art integer

Exercise 12.2 | Q 3. (iii) | Page 248

Determine the truth value of the following statement.

It is not true that 5 + 5 = 9 or Earth is a planet

Exercise 12.2 | Q 3. (iv) | Page 248

Determine the truth value of the following statement.

11 is a prime number and all the sides of a rectangle are equal

Exercise 12.2 | Q 4. (i) | Page 248

Which one of the following sentences is a proposition?

4 + 7 = 12

• is a proposition

• not a proposition

Exercise 12.2 | Q 4. (ii) | Page 248

Which one of the following sentences is a proposition?

What are you doing?

• is a proposition

• not a proposition

Exercise 12.2 | Q 4. (iii) | Page 248

Which one of the following sentences is a proposition?

3n ≤ 81, n ∈ N

• is a proposition

• not a proposition

Exercise 12.2 | Q 4. (iv) | Page 248

Which one of the following sentences is a proposition?

Peacock is our national bird

• is a proposition

• not a proposition

Exercise 12.2 | Q 4. (v) | Page 248

Which one of the following sentences is a proposition?

How tall this mountain is!

• is a proposition

• not a proposition

Exercise 12.2 | Q 5. (i) | Page 248

Write the converse, inverse, and contrapositive of the following implication.

If x and y are numbers such that x = y, then x2 = y2

Exercise 12.2 | Q 5. (ii) | Page 248

Write the converse, inverse, and contrapositive of the following implication.

If a quadrilateral is a square then it is a rectangle

Exercise 12.2 | Q 6. (i) | Page 248

Construct the truth table for the following statement.

¬P ∧ ¬q

Exercise 12.2 | Q 6. (ii) | Page 248

Construct the truth table for the following statement.

¬(P ∧ ¬q)

Exercise 12.2 | Q 6. (iii) | Page 248

Construct the truth table for the following statement.

(p v q) v ¬q

Exercise 12.2 | Q 6. (iv) | Page 248

Construct the truth table for the following statement

(¬p → r) ∧ (p ↔ q)

Exercise 12.2 | Q 7. (i) | Page 249

Verify whether the following compound propositions are tautologies or contradictions or contingency.

(p ∧ q) ∧¬ (p v q)

Exercise 12.2 | Q 7. (ii) | Page 249

Verify whether the following compound propositions are tautologies or contradictions or contingency.

((p v q) ∧¬p) → q

Exercise 12.2 | Q 7. (iii) | Page 249

Verify whether the following compound propositions are tautologies or contradictions or contingency.

(p → q) ↔ (¬p → q)

Exercise 12.2 | Q 7. (iv) | Page 249

Verify whether the following compound propositions are tautologies or contradictions or contingency.

((p → q) ∧ (q → r)) → (p → r)

Exercise 12.2 | Q 8. (i) | Page 249

Show that (p ∧ q) ≡ ¬p v ¬q

Exercise 12.2 | Q 8. (ii) | Page 249

Show that ¬(p → q) ≡ p ∧¬q

Exercise 12.2 | Q 9 | Page 249

Prove that q → p ≡ ¬p → ¬q

Exercise 12.2 | Q 10 | Page 249

Show that p → q and q → p are not equivalent

Exercise 12.2 | Q 11 | Page 249

Show that ¬(p ↔ q) ≡ p ↔ ¬q

Exercise 12.2 | Q 12 | Page 249

Check whether the statement p → (q → p) is a tautology or a contradiction without using the truth table

Exercise 12.2 | Q 13 | Page 249

Using the truth table check whether the statements ¬(p v q) v (¬p ∧ q) and ¬p are logically equivalent

Exercise 12.2 | Q 14 | Page 249

Prove p → (q → r) ≡ (p ∧ q) → r without using the truth table

Exercise 12.2 | Q 15 | Page 249

Prove that p → (¬q v r) ≡ ¬p v (¬q v r) using truth table

Exercise 12.3 [Pages 249 - 251]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 12 Discrete MathematicsExercise 12.3 [Pages 249 - 251]

#### MCQ

Exercise 12.3 | Q 1 | Page 249

Choose the correct alternative:

A binary operation on a set S is a function from

• S → S

• (S × S) → S

• S → (S × S)

• (S × S) → (S × S)

Exercise 12.3 | Q 2 | Page 249

Choose the correct alternative:

Subtraction is not a binary operation in

• R

• Z

• N

• Q

Exercise 12.3 | Q 3 | Page 249

Choose the correct alternative:

Which one of the following is a binary operation on N?

• Subtraction

• Multiplication

• Division

• All the above

Exercise 12.3 | Q 4 | Page 249

Choose the correct alternative:

In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R?

• a * b = min(a.b)

• a * b = max(a, b)

• a * b = a

• a * b = ab

Exercise 12.3 | Q 5 | Page 249

Choose the correct alternative:

The operation * defined by a * b = "ab"/7 is not a binary operation on

• Q+

• Z

• R

• C

Exercise 12.3 | Q 6 | Page 249

Choose the correct alternative:

In the set Q define a ⨀ b = a + b + ab. For what value of y, 3 ⨀ (y ⨀ 5) = 7?

• y = 2/3

• y = (-2)/3

• y = (-2)/3

• y = 4

Exercise 12.3 | Q 7 | Page 249

Choose the correct alternative:

If a * b = sqrt("a"^2 + "b"^2) on the real numbers then * is

• Commutative but not associative

• Associative but not commutative

• Both commutative and associative

• Neither commutative nor associative

Exercise 12.3 | Q 8 | Page 250

Choose the correct alternative:

Which one of the following statements has the truth value T?

• sin x is an even function

• Every square matrix is non-singular

• The product of complex number and its conjugate is purely imaginary

• sqrt(5) is an irrational number

Exercise 12.3 | Q 9 | Page 250

Choose the correct alternative:

Which one of the following statements has truth value F?

• Chennai is in India or sqrt(2) is an integer

• Chennai is in India or sqrt(2) is an irrational number

• Chennai is in China or sqrt(2) is an integer

• Chennai is in China or sqrt(2) is an irrational number

Exercise 12.3 | Q 10 | Page 250

Choose the correct alternative:

If a compound statement involves 3 simple statements, then the number of rows in the truth table is

• 9

• 8

• 6

• 3

Exercise 12.3 | Q 11 | Page 250

Choose the correct alternative:

Which one is the inverse of the statement (p v q) → (p ∧ q)?

• (p ∧ q) → (p v q)

• ¬(p v q) → (p ∧ q

• (¬P v ¬q) → (¬p ∧ ¬q)

• (¬p ∧ ¬q) → (¬p v ¬q)

Exercise 12.3 | Q 12 | Page 250

Choose the correct alternative:

Which one is the contrapositive of the statement (p v q) → r?

• ¬r → (¬p ∧ ¬q)

• ¬r → (p v q)

• r → (p ∧ q)

• p → (q v r)

Exercise 12.3 | Q 13 | Page 250

Choose the correct alternative:

The truth table for (p ∧ q) v ¬q is given below

 p q (p ∧ q) v ¬q T T (a) T F (b) F T (c) F F (d)

Which one of the following is true?

•  (a) T T T T
•  (b) T F T T
•  (c) T T F T
•  (d) T F F F
Exercise 12.3 | Q 14 | Page 250

Choose the correct alternative:

In the last column of the truth table for ¬(p v ¬q) the number of final outcomes of the truth value ‘F’ is

• 1

• 2

• 3

• 4

Exercise 12.3 | Q 15 | Page 251

Choose the correct alternative:

Which one of the following is incorrect? For any two propositions p and q, we have

• ¬(p v q) ≡ ¬p ∧ ¬q

• ¬(p ∧ q) ≡ ¬p v ¬q

• ¬(p v q) ≡ ¬p v ¬q

• ¬(¬p) ≡ p

Exercise 12.3 | Q 16 | Page 251

Choose the correct alternative:

 p q (p ∧ q) → ¬p T T (a) T F (b) F T (c) F F (d)

Which one of the following is correct for the truth value of (p ∧ q) → ¬p

•  (a) T T T T
•  (b) F T T T
•  (c) F F T T
•  (d) T T T F
Exercise 12.3 | Q 17 | Page 251

Choose the correct alternative:

The dual of ¬(p v q) v [p v(p ∧ ¬r)] is

• ¬(p ∧ q) ∧ [p v(p ∧ ¬r)]

• (p ∧ q) ∧ [p v(p v ¬r)]

• ¬(p ∧ q) ∧ [p ∧ (p ∧ r)]

• ¬(p ∧ q) ∧ [p ∧ (p v ¬r)]

Exercise 12.3 | Q 18 | Page 251

Choose the correct alternative:

The proposition p∧(¬p∨q)] is

• a tautology

• logically equivalent to p∧q

• logically equivalent to p∨q

Exercise 12.3 | Q 19 | Page 251

Choose the correct alternative:

Determine the truth value of each of the following statements:
(a) 4 + 2 = 5 and 6 + 3 = 9
(b) 3 + 2 = 5 and 6 + 1 = 7
(c) 4 + 5 = 9 and 1 + 2 = 4
(d) 3 + 2 = 5 and 4 + 7 = 11

•  (a) F T F T
•  (b) T F T F
•  (c) T T F F
•  (D) F F T T
Exercise 12.3 | Q 20 | Page 251

Choose the correct alternative:

Which one of the following is not true?

• Negation of a statement is the statement itself

• If the last column of the truth table contains only T then it is a tautology

• If the last column of its truth table contains only F then it is a contradiction

• If p and q are any two statements then p ⟷ q is a tautology

## Chapter 12: Discrete Mathematics

Exercise 12.1Exercise 12.2Exercise 12.3

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 12 - Discrete Mathematics

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 12 Discrete Mathematics are Concept of Binary Operations, Mathematical Logic, Introduction to Discrete Mathematics.

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