Propositional Logic MCQ Quiz in मल्याळम - Objective Question with Answer for Propositional Logic - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക
Last updated on Mar 15, 2025
Latest Propositional Logic MCQ Objective Questions
Top Propositional Logic MCQ Objective Questions
Propositional Logic Question 1:
The following propositional statement is
[(p → r) ∧ (q → r)] → [(p ∨ q) → r]Answer (Detailed Solution Below)
Propositional Logic Question 1 Detailed Solution
[(p → r) ∧ (q → r)] → [(p ∨ q) → r]
≡ [(p̅ + r).(q̅ +r)] →[ \(\overline{p+q}\) + r] .....p → r =p̅ + r
≡ [(p̅.q̅ + p̅r + rq̅ + r] →[ p̅.q̅ + r]
≡ [(p̅.q̅ + r(p̅ + q̅ + 1)] →[ p̅.q̅ + r]
≡ [(p̅.q̅ + r] →[ p̅.q̅ + r]
≡ True (T → T = T or F → F = T)
The following propositional statement is tautology
Important Point:
For each combination of truth value of p,q,r given expression produce always true value therefore it is tautology. Hence option b is correct.
Propositional Logic Question 2:
Let L (x, y) be the statement "x loves y" where the domain for both x and y consists of all people in the world. Use quantifiers to express "Joy is loved by everyone".
Answer (Detailed Solution Below)
Propositional Logic Question 2 Detailed Solution
The correct answer is ∀x L(x, Joy)
Explanation:
To express the statement "Joy is loved by everyone" using quantifiers and the given predicate L(x, y) (where L(x, y) means "x loves y"), we want to convey that for every person x , that person loves Joy.
This can be expressed using the universal quantifier as follows: 1) ∀x L(x, Joy)
This translates to: "For every person x , x loves Joy," which accurately captures the intended meaning that everyone loves Joy.
Thus, the correct answer is: 1)∀x L(x, Joy) .
Propositional Logic Question 3:
p is the proposition “ Indian army moves back”
q is the proposition “ Chinese army moves back”
r is the proposition “ There is no war”
Then what is the contrapositive of the statement “ If Indian army moves back and Chinese army moves back, then there is no war”
Answer (Detailed Solution Below)
Propositional Logic Question 3 Detailed Solution
Key Points
If the statement is true, then the contrapositive is also logically true.
|
Statement |
If p, then q. |
| Converse | If q, then p. |
| Inverse | If not p, then not q. |
| Contrapositive | If not q, then not p |
The given statement is,
"If Indian army moves back and Chinese army moves back, then there is no war”
For the compound statement p→q, the contrapositive is ~q→~p.
The given statement is in the form ( p∧q) →r
So contrapositive is ~r→~(p∧q).
Hence the correct answer is ~r→~(p∧q).
Propositional Logic Question 4:
Consider the following two well-formed formulas in prepositional logic.
F1: \(P \Longrightarrow \neg \;P\)
F2: \((P \Longrightarrow \neg \;P) \;\vee\; (\neg \; P \Longrightarrow P)\)
Which of the following statements is correct ?
Answer (Detailed Solution Below)
Propositional Logic Question 4 Detailed Solution
The correct answer is option 1.
CONCEPT:
Valid: If for all the combinations of variables, the expression returns true then it is valid.
Satisfiable: If there exists at least one combination of variable which return true then it is satisfiable
Key Points
F1: It is Satisfiable.
| P | ¬P | P⇒¬P |
| T | F | F |
| F | T | T |
F2: It is valid
| P | ¬P | P⇒¬P | ¬P⇒P | (P⇒¬P)∨(¬P⇒P) |
| T | F | F | T | T |
| F | T | T | F | T |
∴ Hence the correct answer is F1 is Satisfiable, F2 is valid.
Additional Information
- A proposition P is a tautology if it is true under all circumstances. It means it contains only TRUE in the final column of its truth table.
- A statement that is always false is known as a contradiction.
- A statement that can be either true or false depending on the truth values of its variables is called a contingency.
Propositional Logic Question 5:
“If X, then Y unless Z” is represented by which of the following formulae in propositional logic ?
Answer (Detailed Solution Below)
Propositional Logic Question 5 Detailed Solution
The correct answer is option 2.
Key Points
“If X, then Y unless Z” means \(\neg Z \to(X \to Y)\)
\(Z \lor \neg X \lor Y \\\neg X\lor Z \lor Y\)
| \(A). \\ \left( {X \wedge Y} \right) \to \neg \;Z \\ \neg (X \land Y ) \lor \neg Z \\ \neg X \lor \neg Y \lor \neg Z\) | \(B). \\ \left( {X \wedge \neg \;Z} \right) \to Y \\ \neg (X \land \neg Z) \lor Y \\ \neg X \lor Z \lor Y\) | \(C. \\ X \rightarrow (Y \wedge \neg\;Z) \\ \neg X \lor (Y \land \neg Z) \\ \neg X \lor Y \land \neg Z\) | \(D. \\ Y \rightarrow (X \wedge \neg\;Z) \\ Y \lor (X \land \neg Z) \\ Y \lor X \land \neg Z \) |
Hence Option 2 only matches \(\\\neg X\lor Z \lor Y\)
Hence the correct answer is \(\left( {X \wedge \neg \;Z} \right) \to Y\)
Propositional Logic Question 6:
Consider P and Q be the two propositions. Which of the following is/are equivalent to ¬ (P ↔ Q).
Answer (Detailed Solution Below)
Propositional Logic Question 6 Detailed Solution
Formula:
¬ (P ↔ Q) = (¬P ↔ Q) = (P ↔ ¬Q)
Truth Table:
|
P |
¬ P |
Q |
¬ Q |
¬ (P ↔ Q) |
P ↔ ¬ Q |
P ↔ ¬ Q |
¬ P ↔ ¬ Q |
Q → P |
|
F |
T |
F |
T |
F |
F |
F |
T |
T |
|
F |
T |
T |
F |
T |
T |
T |
F |
F |
|
T |
F |
F |
T |
T |
T |
T |
F |
T |
|
T |
F |
T |
F |
F |
F |
F |
T |
T |
Therefore, ¬ (P ↔ Q) is equivalent to (¬P ↔ Q) and (P ↔ ¬Q)
Option 1 and 3 are correct
Important Points:
(P ↔ Q) ≡ p ⊙ q
¬ (P ↔ Q) ≡ p ⊕ qPropositional Logic Question 7:
Consider two boolean variables p and r. Which of the following logical expressions is/are tautology?
Answer (Detailed Solution Below)
Propositional Logic Question 7 Detailed Solution
Option 1: TRUE
p → (q → p)
≡ p → (¬ q ∨ p)
≡ ¬ p ∨ (¬ q ∨ p)
≡ (¬ p ∨ ¬ q) ∨ (¬ p ∨ p)
≡ (¬ p ∨ ¬ q) ∨ T
≡ T
Option 2: TRUE
(p \(\wedge\) q) \(\rightarrow\) p
≡ ¬(p \(\wedge\) q) \(\vee\) p
≡ ¬p \(\vee\) ¬q \(\vee\) p
≡ ¬p \(\vee\) p \(\vee\) ¬q
≡ T \(\vee\) ¬q
≡ T
Option 3: False.
¬(p \(\vee\) ¬(p \(\wedge\) q))
= ¬(p \(\vee\) ¬p \(\vee\) ¬q)
= ¬(T \(\vee\) ¬q)
= ¬(T)
= F
Option 4: True.
((p \(\vee\) q) \(\wedge\) ¬p) \(\rightarrow\) q
≡ ¬((p \(\vee\) q) \(\wedge\) ¬p) \(\vee\) q
≡ ¬(p \(\vee\) q) \(\vee\) ( p \(\vee\) q)
≡ T
Propositional Logic Question 8:
In mathematical logic, which of the following are statements?
(i) There will be snow in January
(ii) What is the time now?
(iii) Today is Sunday
(iv) You must study Discrete Mathematics.Answer (Detailed Solution Below)
Propositional Logic Question 8 Detailed Solution
Concept:
In mathematical logic, a statement is a sentence which is either true or false. It may contain words or symbols. In general, it has two parts: 1) assumption 2) conclusion.
A mathematical sentence is a sentence that states a fact and it becomes statement when it results in either true or false.
Explanation:
Statement (i): There will be snow in January
It is a mathematical statement because either there will be snow in January or there will not be snow in January. It has only two possible values either true or false.
Statement (ii): What is the time now?
In this case, there is no meaning of true or false. It is only asking the current time and you can answer that but not in true or false.
Statement (iii): Today is Sunday
It is a statement. As, it is true of Sunday and false on any other day. So, there are two possibilities of either true or false.
Statement (iv): You must study Discrete Mathematics.
It is not a statement. It is not giving meaning in true or false sense.Propositional Logic Question 9:
If a binary operation # is defined as follows
|
A |
B |
A # B |
|
T |
T |
F |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
F |
Answer (Detailed Solution Below)
Propositional Logic Question 9 Detailed Solution
From the Table: A # B = ¬ A ∧ B
- ¬ A # ¬ B = ¬ (¬ A) ∧ ¬ B = A ∧ ¬ B
- A # ¬ B = ¬ A ∧ ¬ (¬ B) = ¬ A ∧ B
- ¬ A # B = ¬ (¬ A) ∧ B = A ∧ B
- A # B = ¬ A ∧ B
Propositional Logic Question 10:
P: The population of Hyderabad is more than Delhi.
Q: Last year, the number of months with 31 days are 7.
choose the correct option based on the truth value of the statements?
Answer (Detailed Solution Below)
Propositional Logic Question 10 Detailed Solution
Option 1: P→Q is True
True, In general, P→ Q is false, only when P is true and Q is false.
If q is true, then irrespective of the P value, it holds true.
The population of Hyderabad is more than Delhi we can not predict which city is more polluted. So it may be True or False.
Here, ‘Q’ is true, as Jan, Mar, May, Jul, Aug, Oct, Dec months have 31 days.
P→Q is True.
Option 2: (P→Q)→ Q is False
False, As Q is true, P→Q and (P→Q )→Q holds true.
Option 3: Q→ (P→Q ) is False
False, As P→Q is true, Q→ ( P→Q) holds true.
Option 4: ¬(P→Q) is True
True, This complement of (P→Q). As (P→Q) is true then ¬(P→Q) is False.
Hence the correct answer is P→Q is True