DI DS LR Test 16

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

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

2 Mark(s)

P P
¬P ¬P
True True
False False

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

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

2 Mark(s)

P P
¬P ¬P
True True
False False

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)

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

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)

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

2 Mark(s)

P P
Q Q
P ∧ Q P ∧ Q
P ∨ Q P ∨ Q

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

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)

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)

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)

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)

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

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

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

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 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

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

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: "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

DI DS LR Test 16 : Question 1 of 25