#### Chapters

Chapter 2: Complex Numbers

Chapter 3: Theory of Equations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Two Dimensional Analytical Geometry-II

Chapter 6: Applications of Vector Algebra

Chapter 7: Applications of Differential Calculus

Chapter 8: Differentials and Partial Derivatives

Chapter 9: Applications of Integration

Chapter 10: Ordinary Differential Equations

Chapter 11: Probability Distributions

Chapter 12: Discrete Mathematics

## Chapter 12: Discrete Mathematics

### 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]

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

a * b – a.|b| on R

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

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

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

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

On Z, define * by (m * n) = m^{n} + n^{m }: ∀m, n ∈ Z Is * binary on Z?

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

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

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.

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.

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 |

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?

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

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

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

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

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

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

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

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

### 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]

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

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

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

p → ¬q

p ↔ q

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

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

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

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

19 is not a prime number

Determine the truth value of the following statement.

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

Determine the truth value of the following statement.

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

Determine the truth value of the following statement.

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

Determine the truth value of the following statement.

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

Which one of the following sentences is a proposition?

4 + 7 = 12

is a proposition

not a proposition

Which one of the following sentences is a proposition?

What are you doing?

is a proposition

not a proposition

Which one of the following sentences is a proposition?

3^{n} ≤ 81, n ∈ N

is a proposition

not a proposition

Which one of the following sentences is a proposition?

Peacock is our national bird

is a proposition

not a proposition

Which one of the following sentences is a proposition?

How tall this mountain is!

is a proposition

not a proposition

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

If x and y are numbers such that x = y, then x^{2} = y^{2}

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

If a quadrilateral is a square then it is a rectangle

Construct the truth table for the following statement.

¬P ∧ ¬q

Construct the truth table for the following statement.

¬(P ∧ ¬q)

Construct the truth table for the following statement.

(p v q) v ¬q

Construct the truth table for the following statement

(¬p → r) ∧ (p ↔ q)

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

(p ∧ q) ∧¬ (p v q)

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

((p v q) ∧¬p) → q

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

(p → q) ↔ (¬p → q)

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

((p → q) ∧ (q → r)) → (p → r)

Show that (p ∧ q) ≡ ¬p v ¬q

Show that ¬(p → q) ≡ p ∧¬q

Prove that q → p ≡ ¬p → ¬q

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

Show that ¬(p ↔ q) ≡ p ↔ ¬q

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

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

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

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

### 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

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)

Choose the correct alternative:

Subtraction is not a binary operation in

R

Z

N

Q

Choose the correct alternative:

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

Subtraction

Multiplication

Division

All the above

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 = a

^{b}

Choose the correct alternative:

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

Q

^{+}Z

R

C

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

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

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

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

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

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)

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)

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

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

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

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

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)]

Choose the correct alternative:

The proposition p∧(¬p∨q)] is

a tautology

a contradiction

logically equivalent to p∧q

logically equivalent to p∨q

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

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

## 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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