Tautology MCQ Quiz in मराठी - Objective Question with Answer for Tautology - मोफत PDF डाउनलोड करा
Last updated on Mar 17, 2025
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Tautology Question 1:
Which one of the following Boolean expression is a tautology?
Answer (Detailed Solution Below)
Tautology Question 1 Detailed Solution
Not tautology (Take both \(p\) and \(q\) as \(T\))
(2) \((p \wedge q) \vee (p\wedge \sim q) \equiv p \wedge (q \vee \sim q) \equiv p \wedge t \equiv p\)
(3) \((p \wedge q) \wedge (p\vee \sim q) \equiv p \vee (q \wedge \sim q) \equiv p \vee c \equiv p\)
(4) \((p \vee q) \vee (p\vee \sim q) \equiv p \vee (q \vee \sim q) \equiv p \vee t \equiv t\)
Tautology Question 2:
For any two statements p and q, the negation of the expression \(p \vee (\sim p \wedge q)\) is?
Answer (Detailed Solution Below)
Tautology Question 2 Detailed Solution
\(\sim (p \vee (\sim p \wedge q))\)
\(= \sim p \wedge \sim (\sim p \wedge q)\)
\(= \sim p \wedge (p \vee \sim q)\)
\(= (\sim p \wedge p) \vee (\sim p \wedge \sim q)\)
\(= F \vee (\sim p \wedge \sim q)\)
\(= (\sim p \wedge \sim q)\)
Tautology Question 3:
The expression \(\sim(\sim p \rightarrow q)\) is logically equivalent to
Answer (Detailed Solution Below)
Tautology Question 3 Detailed Solution
\(p\) \(q\) \(\sim p\) \(\sim p \to q\) \(\sim(\sim p \to q)\) \((\sim p \land \sim q)\)
T F T F T
T T F F F
F T F T T
F F T T F
Tautology Question 4:
The Boolean expression \(\left( p \wedge q \right) \vee \left( p \vee \sim q \right) \wedge \left( \sim p \wedge \sim q \right)\) is equivalent to :
Answer (Detailed Solution Below)
Tautology Question 4 Detailed Solution
Tautology Question 5:
The given circuit is equivalent to
Answer (Detailed Solution Below)
Tautology Question 5 Detailed Solution
Answer : 4
Solution :
The symbolic form of the given circuit is
(p ∨ ~ q∨ ~ r) (p ∨ (q ∧ r))
≡ p ∨ [(~ q∨ ~ r) ∧ (q ∧ r)] ...[Distributive law]
≡ p ∨ [~ (q ∧ r) ∧ (q Ʌ r)] ...[De Morgan's law]
≡ p ∨ F ....[Complement law]
≡ p ...[Identity law]
Tautology Question 6:
The logical statement [~(~p ∨ q) ∨ (p ∧ r)] ∧ (~ q ∧ r) is equivalent to
Answer (Detailed Solution Below)
Tautology Question 6 Detailed Solution
Answer : 2
Solution :
[~(~ p ∨ q) ∨ (p ∧ r)] ∧ (~ q ∧ r)
≡ [(p ∧ ~ q) ∨ (p ∧ r)] ∧ (~ q ∧ r)...[De Morgan's law]
≡ p ∧ (~ q ∨ r) ∧ (~ q ∧ r) ...[Distributive law]
≡ p ∧ [(~ q ∨ r) ∧ ~ g] ∧ r ...[Associative law]
≡ p ∧ (~ q) ∧ r.... [Absorption law]
≡ (p ∧ r) ∧ ~ q...Commutative law]
Tautology Question 7:
For the statements p and q, consider the following compound statements:
(a) (~q∧(p → q)) → ~p
(b) ((p∨q))∧~p) → q
Then which of the following statements is correct?
Answer (Detailed Solution Below)
Tautology Question 7 Detailed Solution
Concept:
Tautology is a formula or assertion that is true in every possible interpretation
Calculation:
Given, (~q∧(p → q)) → ~p
∴ (a) is tautology.
((p∨q))∧~p) → q
∴ (b) is tautology.
∴ (a) and (b) both are tautologies.
The correct answer is Option 3.
Tautology Question 8:
Which of the following is logically equivalent to ~ (p ⇔ q)
Answer (Detailed Solution Below)
Tautology Question 8 Detailed Solution
Calculation:
We know that p ⇔ q = (p ⇒ q) ∧ (q ⇒ p)
= (~ p ∨ q) ∧ (~ q ∨ p)
∴ ~ (p ⇔ q)
= ~(~ p ∨ q) ∨ ~(~ q ∨ p)
= (p ∧ ~q) ∨ (q ∧ ~p)
| p | q | (p ∧ ~q) ∨ (q ∧ ~p) | (~p) ⇔ q | ~p ⇔ ~q | p → ~ q | p → q |
| T | T | F | F | T | F | T |
| T | F | T | T | F | T | F |
| F | T | T | T | F | T | T |
| F | F | F | F | T | T | T |
From truth table, ~ (p ⇔ q) = (~p) ⇔ q
∴ (~p) ⇔ q is logically equivalent to ~ (p ⇔ q).
The correct answer is Option 1.
Tautology Question 9:
Let p and q are two statements then p \(\leftrightarrow \) q is equivalent to
Answer (Detailed Solution Below)
Tautology Question 9 Detailed Solution
Calculation:
Option 1: (p' ∧ q')→q
It means that if both p and q are false, then q is true.
This is not equivalent to p ↔ q.
Option 2: (p' ∧ q') ∨ (p ∧ q)
It means that either both p and q are false, or both p and q are true.
This is exactly what p ↔ q represents.
Option 3: (p' ∨ q') → p
It means that if either p or q is false, then p is true.
This is not equivalent to p ↔ q.
Option 4: (p' ∧ q') ∧ (p ∧ q)
It means that both p and q are false and both p and q are true simultaneously, which is a contradiction.
This is not equivalent to p ↔ q.
∴ p \(\leftrightarrow \) q is equivalent to (p' ∧ q') ∨ (p ∧ q).
The correct answer is Option 2.
Tautology Question 10:
Which of the following statements is a tautology?
Answer (Detailed Solution Below)
Tautology Question 10 Detailed Solution
Concept:
A statement which is always correct is a tautology.
Calculation:
Truth Table
|
p |
q |
~p |
~q |
(~p) ∨ q |
((~p) ∨ q) ⇒ p |
p ⇒ ((~p) ∨ q) |
(~p) ∨ q ⇒ q |
q ⇒ ((~p) ∨ q) |
|
T |
T |
F |
F |
T |
T |
T |
T |
T |
|
T |
F |
F |
T |
F |
T |
F |
T |
T |
|
F |
T |
T |
F |
T |
F |
T |
T |
T |
|
F |
G |
T |
T |
T |
F |
T |
F |
T |
∴ q ⇒ ((~ p) ∨ q) is a tautology.
The correct answer is Option 4.