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Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2

Question 1.
Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 1
Solution:
(i) \(\neg p\) : Jupiter is not a planet
(ii) \(p \wedge \neg q\) : Jupiter is not a planet and India is not an island
(iii) \(\neg p \vee q\) : Jupiter is not a planet or India is an island.
(iv) \(p \rightarrow \neg q\) : If Jupiter is a planet then India is not an island
(v) \(p \leftrightarrow q\) : If Jupiter is a planet if and only if India is an island

Question 2.
Write each of the following sentences in symbolic form using statement variables p and q.
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number
Solution:
p : 19 is a prime number
q : All the angles of a triangle are equal
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 2

Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2

Question 3.
Determine the truth value of each of the following statements
(i) If 6 + 2 = 5 , then the milk is white.
(ii) China is in Europe or \(\sqrt{3}\) is an integer
(iii) It is not true that 5 + 5 = 9 or Earth is a planet
(iv) 11 is a prime number and all the sides of a rectangle are equal
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 3

Question 4.
Which one of the following sentences is a proposition?
(i) 4 + 7 = 12
(ii) What are you doing?
(iii) 3n ≤ 81, n ∈ N
(iv) Peacock is our national bird
(v) How tall this mountain is!
Solution:
(i) is a proposition
(ii) not a proposition
(iii) is a proposition
(iv) is a proposition
(v) not a proposition

Question 5.
Write the converse, inverse, and contrapositive of each of the following implication.
(i) If x and y are numbers such that x = y, then x2 = y2
(ii) If a quadrilateral is a square then it is a rectangle
Solution:
(i) Converse: If x and y are numbers such that x2 = y2 then x = y.
Inverse: If x and y are numbers such that x ≠ y then x2 ≠ y2.
Contrapositive : If x and v are numbers such that x2 ≠ y2 then x ≠ y.

(ii) Converse: If a quadrilateral is a rectangle then it is a square.
Inverse: If a quadrilateral is not a square then it is not a rectangle.
Contrapositive : If a quadrilateral is not a rectangle then it is not a square.

Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2

Question 6.
Construct the truth table for the following statements.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 4
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 5
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 6

Question 7.
Verify whether the following compound propositions are tautologies or contradictions or contingency
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 8
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 9
In the above Truth table the last column entries are ‘F’. So the given propositions is a contradiction.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 10
In the above truth table the last column entries are ‘T’. So the given propositions is a tautology.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 11
In the above truth table the entries in the last column are a combination of’ T ‘ and ‘ F ‘. So the given statement is neither propositions is neither tautology nor a contradiction. It is a contingency.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 12
The last column entires are ‘T’. So the given proposition is a tautology.

Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2

Question 8.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 13
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 14
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 15

Question 9.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 16
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 18
The entries in the column corresponding to q ➝ p and \(\neg p \rightarrow \neg q\) are identical and hence they are equivalent.

Question 10.
Show that p ➝ q and q ➝ p are not equivalent
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 19
The entries in the column corresponding to p ➝ q and q ➝ p are not identical, hence they are not equivalent.

Question 11.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 20
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 21

Question 12.
Check whether the statement p ➝ (q ➝ p) is a tautology or a contradiction without using the truth table.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 22

Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2

Question 13.
Using truth table check whether the statements Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 23 are logically equivalent.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 24

Question 14.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 25 without using truth table
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 26

Question 15.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 27
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 28
The entries in the column corresponding to Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 29 are identical.
Hence they are equivalent.

Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2

Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 Additional Problems

Question 1.
Show that Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 367 is a tautology.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 31

Question 2.
Show that Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 368 is a contradiction.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 369

Question 3.
Use the truth table to determine whether the statement Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 374 is a tautology.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 370
The last column contains only T. ∴ The given statement is a tautology.

Question 4.
Show that Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 371
Solution:
(i) Truth table for p \(\leftrightarrow\) q
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 372
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 373

Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2

Question 5.
Show that Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 36.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 366
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 37
The last columns of statements (i) and (ii) are identical.
Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.2 38

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