Class 12 Mathematics 1 Mathematical Logic Exercise 1.2 Solutions | Maharashtra Board

 

Class 12 Mathematics 1 – Mathematical Logic

Exercise 1.2 – Solutions

Topic: Truth Tables, Logical Equivalences, Tautology, Contradiction and Contingency

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Q.1. Construct the truth table for each of the following statement patterns.

i) [(p → q) ∧ q] → p

Answer: Construct the truth table using the columns p, q, p → q, (p → q) ∧ q and [(p → q) ∧ q] → p.

ii) (p ∧ ~q) ↔ (p → q)

Answer: Construct the truth table using the columns p, q, ~q, p ∧ ~q, p → q and the final biconditional.

iii) (p ∧ q) ↔ (q ∨ r)

Answer: Since three variables are involved, the truth table contains 8 rows.

iv) p → [~(q ∧ r)]

Answer: Since three variables are involved, the truth table contains 8 rows.

v) ~p ∧ [(p ∨ ~q) ∧ q]

Answer: The final statement is always false.

vi) (~p → ~q) ∧ (~q → ~p)

Answer: Construct the truth table using p, q and the two implications.

vii) (q → p) ∨ (~p ↔ q)

Answer: Construct the truth table using p, q and the two component statements.

viii) [p → (q → r)] ↔ [(p ∧ q) → r]

Answer: The final statement is a tautology.

ix) p → [~(q ∧ r)]

Answer: Since three variables are involved, the truth table contains 8 rows.

x) (p ∨ ~q) → (r ∧ p)

Answer: Since three variables are involved, the truth table contains 8 rows.

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Q.2. Using truth tables prove the following logical equivalences.

To prove a logical equivalence, construct the truth table and show that the final columns of both sides have identical truth values.

i)

~p ∧ q ≡ (p ∨ q) ∧ ~p

Result: Proved.

ii)

~(p ∨ q) ∨ (~p ∧ q) ≡ ~p

Result: Proved.

iii)

p ↔ q ≡ ~[(p ∨ q) ∧ ~(p ∧ q)]

Result: Proved.

iv)

p → (q → p) ≡ ~p → (p → q)

Result: Proved.

v)

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

Result: Proved.

vi)

p → (q ∧ r) ≡ (p → q) ∧ (p → r)

Result: Proved.

vii)

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Result: Proved.

viii)

[~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

Result: Proved.

ix)

~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p)

Result: Proved.

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Q.3. Examine whether each of the following statement patterns is a tautology, contradiction or a contingency.

Q. No.	Statement Pattern	Answer
i)	(p ∧ q) → (q ∨ p)	Tautology
ii)	(p → q) ↔ (~p ∨ q)	Tautology
iii)	[~(~p ∧ ~q)] ∨ q	Tautology
iv)	[(p → q) ∧ q] → p	Contingency
v)	[(p → q) ∧ ~q] → ~p	Tautology
vi)	(p ↔ q) ∧ (p → ~q)	Contingency
vii)	~(~q ∧ p) ∧ q	Contingency
viii)	(p ∧ ~q) ↔ (p → q)	Contradiction
ix)	(~p → q) ∧ (p ∧ r)	Contingency
x)	[p → (~q ∨ r)] ↔ ~[p → (q → r)]	Contradiction

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Important Definitions

1. Tautology: A statement which is true for every possible combination of truth values.

2. Contradiction: A statement which is false for every possible combination of truth values.

3. Contingency: A statement which is true for some combinations and false for others.