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### Mathematical Logic Exercise 1.6 [Page 16] Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1

EXERCISE 1.6 [PAGE 16]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.6 [Page 16]

EXERCISE 1.6Q 1.1   PAGE 16
Exercise 1.6 | Q 1.1 | Page 16

Prepare truth tables for the following statement pattern.

p → (~ p ∨ q)

#### SOLUTION

p → (~ p ∨ q)

 p q ~p ~ p ∨ q p → (~ p ∨ q) T T F T T T F F F F F T T T T F F T T T
EXERCISE 1.6Q 1.2   PAGE 16
Exercise 1.6 | Q 1.2 | Page 16

Prepare truth tables for the following statement pattern.

(~ p ∨ q) ∧ (~ p ∨ ~ q)

#### SOLUTION

(~ p ∨ q) ∧ (~ p ∨ ~ q)

 p q ~p ~q ~p∨q ~p∨~q (~p∨q)∧(~p∨~q) T T F F T F F T F F T F T F F T T F T T T F F T T T T T
EXERCISE 1.6Q 1.3   PAGE 16
Exercise 1.6 | Q 1.3 | Page 16

Prepare truth tables for the following statement pattern.

(p ∧ r) → (p ∨ ~ q)

#### SOLUTION

(p ∧ r) → (p ∨ ~ q)

 p q r ~q p ∧ r p∨~q (p ∧ r) → (p ∨ ~ q) T T T F T T T T T F F F T T T F T T T T T T F F T F T T F T T F F F T F T F F F F T F F T T F T T F F F T F T T
EXERCISE 1.6Q 1.4   PAGE 16
Exercise 1.6 | Q 1.4 | Page 16

Prepare truth tables for the following statement pattern.

(p ∧ q) ∨ ~ r

#### SOLUTION

(p ∧ q) ∨ ~ r

 p q r ~r p ∧ q (p ∧ q) ∨ ~ r T T T F T T T T F T T T T F T F F F T F F T F T F T T F F F F T F T F T F F T F F F F F F T F T
EXERCISE 1.6Q 2.1   PAGE 16
Exercise 1.6 | Q 2.1 | Page 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

q ∨ [~ (p ∧ q)]

#### SOLUTION

 p q p ∧ q ~ (p ∧ q) q ∨ [~ (p ∧ q)] T T T F T T F F T T F T F T T F F F T T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6Q 2.2   PAGE 16
Exercise 1.6 | Q 2.2 | Page 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

(~ q ∧ p) ∧ (p ∧ ~ p)

#### SOLUTION

 p q ~p ~q (~q∧p) (p∧~p) (~q∧p)∧(p∧~p) T T F F F F F T F F T T F F F T T F F F F F F T T F F F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 2.3   PAGE 16
Exercise 1.6 | Q 2.3 | Page 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

(p ∧ ~ q) → (~ p ∧ ~ q)

#### SOLUTION

 p q ~p ~q p∧~q ~p∧~q (p∧~q)→(~p∧~q) T T F F F F T T F F T T F F F T T F F F T F F T T F T T

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6Q 2.4   PAGE 16
Exercise 1.6 | Q 2.4 | Page 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

~ p → (p → ~ q)

#### SOLUTION

 p q ~p ~q p→~q ~p→(p→~q) T T F F F T T F F T T T F T T F T T F F T T T T

All the truth values in the last column are T. Hence, it is tautology.

EXERCISE 1.6Q 3.1   PAGE 16
Exercise 1.6 | Q 3.1 | Page 16

Prove that the following statement pattern is a tautology.

(p ∧ q) → q

#### SOLUTION

 p q p ∧ q (p∧q)→q T T T T T F F T F T F T F F F T

All the truth values in the last column are T. Hence, it is tautology.

EXERCISE 1.6Q 3.2   PAGE 16
Exercise 1.6 | Q 3.2 | Page 16

Prove that the following statement pattern is a tautology.

(p → q) ↔ (~ q → ~ p)

#### SOLUTION

 p q ~p ~q p→q ~q→~p (p→q)↔(~q→~p) T T F F T T T T F F T F F T F T T F T T T F F T T T T T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6Q 3.3   PAGE 16
Exercise 1.6 | Q 3.3 | Page 16

Prove that the following statement pattern is a tautology.

(~p ∧ ~q ) → (p → q)

#### SOLUTION

 p q ~p ~q ~p∧~q p→q (~p∧~q)→(p→q) T T F F F T T T F F T F F T F T T F F T T F F T T T T T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6Q 3.4   PAGE 16
Exercise 1.6 | Q 3.4 | Page 16

Prove that the following statement pattern is a tautology.

(~ p ∨ ~ q) ↔ ~ (p ∧ q)

#### SOLUTION

 p q ~p ~q ~p∨~q p∧q ~p∨~q (~p∨~q↔~(p ∧ q) T T F F F T F T T F F T T F T T F T T F T F T T F F T T T F T T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6Q 4.1   PAGE 16
Exercise 1.6 | Q 4.1 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∨ q) ∧ (~p ∧ ~q)

#### SOLUTION

 p q ~p ~q p∨q ~p∧~q (p∨q)∧(~p∧~q) T T F F T F F T F F T T F F F T T F T F F F F T T F T F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 4.2   PAGE 16
Exercise 1.6 | Q 4.2 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∧ q) ∧ ~p

#### SOLUTION

 p q ~p p∧q (p∧q)∧~p T T F T F T F F F F F T T F F F F T F F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 4.3   PAGE 16
Exercise 1.6 | Q 4.3 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∧ q) ∧ (~p ∨ ~q)

#### SOLUTION

 p q ~p ~q p∧q ~p∨~q (p∧q)∧(~p∨~q) T T F F T F F T F F T F T F F T T F F T F F F T T F T F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 4.4   PAGE 16
Exercise 1.6 | Q 4.4 | Page 16

Prove that the following statement pattern is a contradiction.

(p → q) ∧ (p ∧ ~ q)

#### SOLUTION

 p q ~q p→q p∧~q (p→q)∧(p∧~q) T T F T F F T F T F T F F T F T F F F F T T F F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 5.1   PAGE 16
Exercise 1.6 | Q 5.1 | Page 16

Show that the following statement pattern is contingency.

(p∧~q) → (~p∧~q)

#### SOLUTION

 p q ~p ~q p∧~q ~p∧~q (p∧~q)→(~p∧~q) T T F F F F T T F F T T F F F T T F F F T F F T T F T T

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6Q 5.2   PAGE 16
Exercise 1.6 | Q 5.2 | Page 16

Show that the following statement pattern is contingency.

(p → q) ↔ (~ p ∨ q)

#### SOLUTION

 p q ~p p→q ~p∨q (p→q)↔(~p∨q) T T F T T T T F F F F T F T T T T T F F T T T T

All the truth values in the last column are T. Hence, it is a tautology. Not contingency.

EXERCISE 1.6Q 5.3   PAGE 16
Exercise 1.6 | Q 5.3 | Page 16

Show that the following statement pattern is contingency.

p ∧ [(p → ~ q) → q]

#### SOLUTION

 p q ~q p→~q (p→~q)→q p∧[(p→~q)→q] T T F F T T T F T T F F F T F T T F F F T T F F

Truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6Q 5.4   PAGE 16
Exercise 1.6 | Q 5.4 | Page 16

Show that the following statement pattern is contingency.

(p → q) ∧ (p → r)

#### SOLUTION

 p q r p→q p→r (p→q)∧(p→r) T T T T T T T T F T F F T F T F T F T F F F F F F T T T T T F T F T T T F F T T T T F F F T T T

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6Q 6.1   PAGE 16

Exercise 1.6 | Q 6.1 | Page 16

Using the truth table, verify

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

#### SOLUTION

 1 2 3 4 5 6 7 8 p q r q∧r p∨(q∧r) p∨q p∨r (p∨q)∧(p∨r) T T T T T T T T T T F F T T T T T F T F T T T T T F F F T T T T F T T T T T T T F T F F F T F F F F T F F F T F F F F F F F F F

The entries in columns 5 and 8 are identical.

∴ p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

EXERCISE 1.6Q 6.2   PAGE 16
Exercise 1.6 | Q 6.2 | Page 16

Using the truth table, verify

p → (p → q) ≡ ~ q → (p → q)

#### SOLUTION

 1 2 3 4 5 6 p q ~q p→q p→(p→q) ~q→(p→q) T T F T T T T F T F F F F T F T T T F F T T T T

In the above truth table, entries in columns 5 and 6 are identical.

∴ p → (p → q) ≡ ~ q → (p → q)

EXERCISE 1.6Q 6.3   PAGE 16
Exercise 1.6 | Q 6.3 | Page 16

Using the truth table, verify

~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q

#### SOLUTION

 1 2 3 4 5 6 7 8 p q ~q p→~q ~(p→~q) ~(~q) p∧~(~q) p∧q T T F F T T T T T F T T F F F F F T F T F T F F F F T T F F F F

In the above table, entries in columns 5, 7, and 8 are identical.

∴ ~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q

EXERCISE 1.6Q 6.4   PAGE 16
Exercise 1.6 | Q 6.4 | Page 16

Using the truth table, verify

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

#### SOLUTION

 1 2 3 4 5 6 7 p q ~p (p∨q) ~(p∨q) ~p∧q ~(p∨q)∨(~p∧q) T T F T F F F T F F T F F F F T T T F T T F F T F T F T

In the above truth table, the entries in columns 3 and 7 are identical.

∴ ~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

EXERCISE 1.6Q 7.1   PAGE 16

Exercise 1.6 | Q 7.1 | Page 16

Using the truth table, verify

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

#### SOLUTION

 1 2 3 4 5 6 7 8 p q r q∧r p∨(q∧r) p∨q p∨r (p∨q)∧(p∨r) T T T T T T T T T T F F T T T T T F T F T T T T T F F F T T T T F T T T T T T T F T F F F T F F F F T F F F T F F F F F F F F F

The entries in columns 5 and 8 are identical.

∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

EXERCISE 1.6Q 7.2   PAGE 16
Exercise 1.6 | Q 7.2 | Page 16

Prove that the following pair of statement pattern is equivalent.

p ↔ q and (p → q) ∧ (q → p)

#### SOLUTION

 1 2 3 4 5 6 p q p↔q p→q q→p (p→q)∧(q→p) T T T T T T T F F F T F F T F T F F F F T T T T

In the above table, entries in columns 3 and 6 are identical.

∴ Statement p ↔ q and (p → q) ∧ (q → p) are equivalent.

EXERCISE 1.6Q 7.3   PAGE 16
Exercise 1.6 | Q 7.3 | Page 16

Prove that the following pair of statement pattern is equivalent.

p → q and ~ q → ~ p and ~ p ∨ q

#### SOLUTION

 1 2 3 4 5 6 7 p q ~p ~q p→q ~q→~p ~p∨q T T F F T T T T F F T F F F F T T F T T T F F T T T T T

In the above table, entries in columns 5, 6 and 7 are identical.

∴ Statement p → q and ~q → ~p and ~p ∨ q are equivalent.

EXERCISE 1.6Q 7.4   PAGE 16
Exercise 1.6 | Q 7.4 | Page 16

Prove that the following pair of statement pattern is equivalent.

~(p ∧ q) and ~p ∨ ~q

#### SOLUTION

 1 2 3 4 5 6 7 p q ~p ~q p∧q ~(p∧q) ~p∨~q T T F F T F F T F F T F T T F T T F F T T F F T T F T T

In the above table, entries in columns 6 and 7 are identical.

∴ Statement ~(p ∧ q) and ~p ∨ ~q are equivalent.