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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.

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

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:

Question 4.

Which one of the following sentences is a proposition?

(i) 4 + 7 = 12

(ii) What are you doing?

(iii) 3^{n} ≤ 81, n ∈ N

(iv) Peacock is our national bird

(v) How tall this mountain is!

Solution:

(i) 4 + 7 = 12

It is a proposition as its truth value is F

(ii) What are you doing?

It is a question and not a proposition.

(iii) 3ⁿ ≤ 81, n ∈ N

It is a proposition as it is true when,

n = 1, 2, 3, 4

(iv) Peacock is our national bird. It is a proposition as its truth value is T.

(v) How tall this mountain is!

This is an exclamation, 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 x^{2} = y^{2}

(ii) If a quadrilateral is a square then it is a rectangle

Solution:

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

Convene: 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 y are numbers such that x² ≠ y² then x ≠ y

(ii) If a quadrilateral is a square then it is a rectangle

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.

Question 6.

Construct the truth table for the following statements.

Solution:

Question 7.

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

Solution:

In the above Truth table the last column entries are ‘F’. So the given propositions is a contradiction.

In the above truth table, the last column entries are ‘T’. So the given proposition is a tautology.

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.

The last column entires are ‘T’. So the given proposition is a tautology.

Question 8.

Solution:

Question 9.

Solution:

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:

The entries in the column corresponding to p ➝ q and q ➝ p are not identical, hence they are not equivalent.

Question 11.

Solution:

Question 12.

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

Solution:

Question 13.

Using the truth table check whether the statements are logically equivalent.

Solution:

Question 14.

without using truth table

Solution:

Question 15.

Solution:

The entries in the column corresponding to are identical.

Hence they are equivalent.

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

Question 1.

Show that is a tautology.

Solution:

Question 2.

Show that is a contradiction.

Solution:

Question 3.

Use the truth table to determine whether the statement is a tautology.

Solution:

The last column contains only T. ∴ The given statement is a tautology.

Question 4.

Show that

Solution:

(i) Truth table for p \(\leftrightarrow\) q

Question 5.

Show that .

Solution:

The last columns of statements (i) and (ii) are identical.