DI DS LR Test 16

Simplify: (P ∧ Q) ∨ (¬P ∧ R) ∨ (Q ∧ R) Simplify: (P ∧ Q) ∨ (¬P ∧ R) ∨ (Q ∧ R)

2 Mark(s)

(P ∧ Q) ∨ (¬P ∧ R) (P ∧ Q) ∨ (¬P ∧ R)
(Q ∨ R) (Q ∨ R)
(P ∨ R) ∧ (Q ∨ R) (P ∨ R) ∧ (Q ∨ R)
(P ∨ Q) ∧ (¬P ∨ R) (P ∨ Q) ∧ (¬P ∨ R)

A says: "B is a knight." B says: "At most one of us is a knight." Determine a consistent assignment. A says: "B is a knight." B says: "At most one of us is a knight." Determine a consistent assignment.

2 Mark(s)

A knight, B knave A knight, B knave
A knave, B knight A knave, B knight
Both knights Both knights
Both knaves Both knaves

A says: "B is a knave." B says: "C is a knave." C says: "A is a knave." Exactly one is a knight. Who? A says: "B is a knave." B says: "C is a knave." C says: "A is a knave." Exactly one is a knight. Who?

2 Mark(s)

A A
B B
C C
None None

Simplify: (P ∧ P) ∨ (P ∧ ¬P) Simplify: (P ∧ P) ∨ (P ∧ ¬P)

2 Mark(s)

P P
¬P ¬P
True True
False False

XNOR (equivalence) P ⊙ Q is true when: XNOR (equivalence) P ⊙ Q is true when:

2 Mark(s)

P and Q have same truth value P and Q have same truth value
Exactly one is true Exactly one is true
At least one is false At least one is false
Both false only Both false only

If P → Q and Q → R are true, which must be true? If P → Q and Q → R are true, which must be true?

2 Mark(s)

P → R P → R
R → P R → P
P ↔ R P ↔ R
P ∧ R P ∧ R

Which inference is invalid? Which inference is invalid?

2 Mark(s)

If P then Q; ¬Q; therefore ¬P (modus tollens)
If P then Q; Q; therefore P (affirming consequent)
P or Q; ¬P; therefore Q (disjunctive syllogism)
P; therefore P ∨ Q (addition) P; therefore P ∨ Q (addition)

A Boolean function of 3 variables has how many rows in its full truth table? A Boolean function of 3 variables has how many rows in its full truth table?

2 Mark(s)

8 8
6 6
4 4
16 16

The contrapositive of "If P then Q" is: The contrapositive of "If P then Q" is:

2 Mark(s)

If ¬Q then ¬P If ¬Q then ¬P
If ¬P then ¬Q If ¬P then ¬Q
If Q then P If Q then P
Q only if P Q only if P

A says: "At least one of B or C is a knight." B says: "C is a knave." Exactly one knight among A,B,C. Who is the knight? A says: "At least one of B or C is a knight." B says: "C is a knave." Exactly one knight among A,B,C. Who is the knight?

2 Mark(s)

A A
B B
C C
None None

Which formula is unsatisfiable? Which formula is unsatisfiable?

2 Mark(s)

P ∧ ¬P P ∧ ¬P
P ∨ ¬P P ∨ ¬P
(P → Q) ∧ P (P → Q) ∧ P
P ↔ Q P ↔ Q

A says: "B is a knave or C is a knave." B says: "A is a knight." If exactly one of A, B, C is a knight, who is it? A says: "B is a knave or C is a knave." B says: "A is a knight." If exactly one of A, B, C is a knight, who is it?

2 Mark(s)

A A
B B
C C
None None

Which is the Disjunctive Normal Form (DNF) of (P ∧ Q) ∨ (P ∧ ¬Q)? Which is the Disjunctive Normal Form (DNF) of (P ∧ Q) ∨ (P ∧ ¬Q)?

2 Mark(s)

P P
Q Q
P ∨ Q P ∨ Q
P ∧ (Q ∨ ¬Q) P ∧ (Q ∨ ¬Q)

How many distinct Boolean functions exist on 2 variables? How many distinct Boolean functions exist on 2 variables?

2 Mark(s)

16 16
8 8
4 4
32 32

Simplify: P ∨ (P ∧ Q) equals: Simplify: P ∨ (P ∧ Q) equals:

2 Mark(s)

P P
Q Q
P ∧ Q P ∧ Q
P ∨ Q P ∨ Q

Is (P ∨ Q) ∧ (¬P ∨ Q) ∧ (P ∨ ¬Q) satisfiable? Is (P ∨ Q) ∧ (¬P ∨ Q) ∧ (P ∨ ¬Q) satisfiable?

2 Mark(s)

Yes; P true, Q arbitrary Yes; P true, Q arbitrary
Yes; P and Q must be true Yes; P and Q must be true
No, unsatisfiable No, unsatisfiable
Yes; Q true, P arbitrary Yes; Q true, P arbitrary

P → Q is equivalent to: P → Q is equivalent to:

2 Mark(s)

¬P ∨ Q ¬P ∨ Q
P ∧ Q P ∧ Q
P ∨ Q P ∨ Q
¬(P ∨ Q) ¬(P ∨ Q)

¬(¬P) is equivalent to: ¬(¬P) is equivalent to:

2 Mark(s)

P P
¬P ¬P
True True
False False

A function in SOP has minterms m(1,3,7). How many input variables if max index is 7? A function in SOP has minterms m(1,3,7). How many input variables if max index is 7?

2 Mark(s)

3 variables 3 variables
2 variables 2 variables
4 variables 4 variables
Cannot say Cannot say

The dual of (P ∨ Q) ∧ R is: The dual of (P ∨ Q) ∧ R is:

2 Mark(s)

(P ∧ Q) ∨ R (P ∧ Q) ∨ R
(P ∧ Q) ∧ R (P ∧ Q) ∧ R
(P ∨ Q) ∨ R (P ∨ Q) ∨ R
(P ∧ Q) ∨ R with constants swapped (P ∧ Q) ∨ R with constants swapped

Which is a tautology? Which is a tautology?

2 Mark(s)

(P → Q) ∨ (Q → P) (P → Q) ∨ (Q → P)
P → (P ∧ Q) P → (P ∧ Q)
P ∧ (P → Q) P ∧ (P → Q)
P ↔ ¬P P ↔ ¬P

For P ∧ (Q ∨ R) to be true, which must hold? For P ∧ (Q ∨ R) to be true, which must hold?

2 Mark(s)

P true and at least one of Q, R true P true and at least one of Q, R true
All P, Q, R true All P, Q, R true
Exactly one of Q, R true Exactly one of Q, R true
P false and Q true P false and Q true

Eliminate → in (P → Q) ∧ (Q → R): Eliminate → in (P → Q) ∧ (Q → R):

2 Mark(s)

(¬P ∨ Q) ∧ (¬Q ∨ R) (¬P ∨ Q) ∧ (¬Q ∨ R)
(P ∨ Q) ∧ (Q ∨ R) (P ∨ Q) ∧ (Q ∨ R)
(P ∧ Q) ∨ (Q ∧ R) (P ∧ Q) ∨ (Q ∧ R)
¬(P ∨ Q ∨ R) ¬(P ∨ Q ∨ R)

In a 2-variable K-map, how many adjacent cells form the largest power-of-two grouping? In a 2-variable K-map, how many adjacent cells form the largest power-of-two grouping?

2 Mark(s)

Which is in Conjunctive Normal Form (CNF)? Which is in Conjunctive Normal Form (CNF)?

2 Mark(s)

(P ∨ Q) ∧ (¬P ∨ R) (P ∨ Q) ∧ (¬P ∨ R)
(P ∧ Q) ∨ R (P ∧ Q) ∨ R
P Q R (concatenated) P Q R (concatenated)
P ↔ Q P ↔ Q

DI DS LR Test 16 : Question 1 of 25