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

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Class 12th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com
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Chapter 12: Discrete Mathematics

Exercise 12.1Exercise 12.2Exercise 12.3
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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

  • a contradiction

  • 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
Class 12th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com

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

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 12 (Discrete Mathematics) 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 Tamil Nadu Board of Secondary Education Class 12th Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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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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Get the free view of chapter 12 Discrete Mathematics Class 12th extra questions for Class 12th Mathematics Volume 1 and 2 Answers Guide and can use Shaalaa.com to keep it handy for your exam preparation

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