Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 12 Discrete Mathematics Ex 12.3 Text Book Back Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.3

Choose the most suitable answer from the given four alternatives:

Question 1.

A binary operation on a set Sis a function from

(a) S → S

(b) (S × S) → S

(c) S → (S × S)

(d) (S × S) → (S × S)

Solution:

(b) (S × S) → S

Question 2.

Subtraction is not a binary operation in

(a) R

(b) Z

(c) N

(d) Q

Solution:

(c) N

Hint:

Let 1, 2 ∈ N

1 * 2 = 1 – 2 = -1 ∉ N

Question 3.

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

(a) Subtraction

(b) Multiplication

(c) Division

(d) All the above

Solution:

(b) Multiplication

Question 4.

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

(a) a * b = min (a.b)

(b) a * b = max (a, b)

(c) a * b = a

(d) a * b = a^{b}

Solution:

(d) a * b = a^{b}

Hint:

Since -2, 1/2 ∈ R , but (-2)^{1/2} ∉ R.

Question 5.

The operation * defined by a * b = \(\frac { ab }{ 7 }\) is not a binary operation on

(a) Q^{+}

(b) Z

(c) R

(d) C

Solution:

(b) Z

Hint:

a * b = \(\frac { ab }{ 7 }\) ∉ Z

as Z is set of all integers.

Question 6.

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

(a) y = \(\frac { 2 }{ 3 }\)

(b) y = \(\frac { -2 }{ 3 }\)

(c) y = \(\frac { -3 }{ 2 }\)

(d) y = 4

Solution:

(b) y = \(\frac { -2 }{ 3 }\)

Hint:

a \(\bigodot\) b = a + b + ab

Given 3 \(\bigodot\) (y \(\bigodot\) 5) = 7

3 \(\bigodot\) (y + 5 + 5y) = 7

3 \(\bigodot\) (6y + 5) = 7

3 + 6y + 5 + 3 (6y + 5) = 7

8 + 6y + 18y + 15 = 7

24 y = 7 – 23 = -16

y = \(\frac { -16 }{ 24 }\) = \(\frac { -2 }{ 3 }\)

Question 7.

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

(a) Commutative but not associative

(b) Associative but not commutative

(c) Both commutative and associative

(d) Neither commutative nor associative

Solution:

(c) Both commutative and associative

Hint:

∴ (a * b) * c = a * (b * c)

* is associative

Question 8.

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

(a) sin x is an even function.

(b) Every square matrix is non-singular

(c) The product of complex number and its conjugate is purely imaginary

(d) \(\sqrt{5}\) is an irrational number

Solution:

(d) \(\sqrt{5}\) is an irrational number

Question 9.

Which one of the following statements has truth value F?

(a) Chennai is in India or √2 is an integer.

(b) Chennai is in India or √2 is an irrational number.

(c) Chennai is in China or √2 is an integer.

(d) Chennai is in China or √2 is an irrational number.

Solution:

(c) Chennai is in China or √2 is an integer.

Question 10.

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

(a) 9

(b) 8

(c) 6

(d) 3

Solution:

(b) 8

Hint:

(i.e.) 2^{3} = 8

Question 11.

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

(a) (p ∧ q) → (p v q)

(b) ¬(p v q) → (p ∧ q)

(c) (¬P v ¬q) → (¬p ∧ ¬q)

(d) (¬p ∧ ¬q) → (¬p v ¬q)

Solution:

(d) (¬p ∧ ¬q) → (¬p v ¬q)

Hint:

(p v q) → (p ∧ q)

¬(p v q) → ¬(p ∧ q)

(¬P ∧ ¬q) → (¬p v ¬q)

Question 12.

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

(a) ¬r → (¬p ∧ ¬q)

(b) ¬r → (p v q)

(c) r → (p ∧ q)

(d) p → (q v r)

Solution:

(a) ¬r → (¬p ∧ ¬q)

Hint:

(p v q) → r

Contrapositive is ¬r → ¬(p v q)

¬r → (¬p ∧ ¬q)

Question 13.

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

Solution:

(c) (1) T, (2) T, (3) F, (4) T

Hint:

1 = T, 2 = T, 3 = F, 4 = T

Question 14.

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

(a) 1

(b) 2

(c) 3

(d) 4

Solution:

(c) 3

Hint:

Number of ‘F’ in last column is 3

Question 15.

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

(a) ¬(p v q) ≡ ¬p ∧ ¬q

(b) ¬(p ∧ q) ≡ ¬p v ¬q

(c) ¬(p v q) ≡ ¬p v ¬q

(d) ¬(¬p) ≡ p

Solution:

(c) ¬(p v q) ≡ ¬p v ¬q

Question 16.

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

Solution:

(b) (1) F, (2) T, (3) T, (4) T

Hint:

Question 17.

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

(a) ¬(p ∧ q) ∧ [p v(p ∧ ¬r)]

(b) (p ∧ q) ∧ [p v(p v ¬r)]

(c) ¬(p ∧ q) ∧ [p ∧ (p ∧ r)]

(d) ¬(p ∧ q) ∧ [p ∧ (p v ¬r)]

Solution:

(d) ¬(p ∧ q) ∧ [p ∧ (p v ¬r)]

Hint:

Dual is obtained by changing ∧ into v and vice versa.

Question 18.

The proposition \(p \wedge(\neg p \vee q)]\) is ……..

(a) a tautology

(b) a contradiction

(c) logically equivalent to \(p \wedge q\)

(d) logically equivalent to \(p \vee q\)

Solution:

(c) logically equivalent to \(p \wedge q\)

Question 19.

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

Solution:

(a) (1) F, (2) T, (3) F, (4) T

Hint:

(1) F and T = F

(2) T and T =T

(3) T and F = F

(4) T and T = T

Question 20.

Which one of the following is not true?

(a) Negation of a statement is the statement itself.

(b) If the last column of the truth table contains only T then it is a tautology.

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

(d) If p and q are any two statements then p ⟷ q is a tautology.

Solution:

(d) If p and q are any two statements then p ⟷ q is a tautology.