Page 48 - Computer science 868 Class 12
P. 48
2. Prove ∼(a → b) ∨ (∼a ∨ (a ∧ b)) is a tautology.
Ans.
a b ∼a a → b ∼(a → b) a ∧ b ∼a∨ (a ∧ b) ∼(a → b) ∨ (∼a∨ (a ∧ b))
0 0 1 1 0 0 1 1
0 1 1 1 0 0 1 1
1 0 0 0 1 0 0 1
1 1 0 1 0 1 1 1
3. If, A = “Taj Mahal is one of the seven wonders of the world.” and
B = “It is a favourite tourist destination.”
Then, write statements for the following propositions.
(i) A ∧ B (ii) A →B (iii) A ↔ B (iv) A' ∨ B' (v) B' → A
Ans. (i) Taj Mahal is one of the seven wonders of the world and it is a favourite tourist destination.
(ii) If, the Taj Mahal is one of the seven wonders of the world then it is a favourite tourist destination.
(iii) Taj Mahal is one of the seven wonders of the world if and only if it is a favourite tourist destination.
(iv) Taj Mahal is not one of the seven wonders of the world or it is not a favourite tourist destination.
(v) If it is not a favourite tourist destination, then the Taj Mahal is one of the seven wonders of the world.
4. Draw the truth table of (A9B) 5 C.
Ans. A B C A 9 B (A 9 B) 5 C
0 0 0 1 1
0 0 1 1 0
0 1 0 0 0
0 1 1 0 1
1 0 0 0 0
1 0 1 0 1
1 1 0 1 1
1 1 1 1 0
5. Prove (A 5 B) 5 C = A 5 (B 5 C) using truth table.
Ans.
A B C A 5 B (A 5 B) 5 C A 9 B (A 9 B) 5 C
0 0 0 0 0 0 0
0 0 1 0 1 1 1
0 1 0 1 1 1 1
0 1 1 1 0 0 0
1 0 0 1 1 0 1
1 0 1 1 0 1 0
1 1 0 0 0 1 0
1 1 1 0 1 0 1
We find that the fifth column and the last column are identical. Hence proved the given equation is true.
6. Minimise the following using Boolean laws:
(a) (A'+C).(A'+C').(A'+B+C'.D) [ISC 2014]
=(A'+ C.C').(A'+ B + C'.D) [Distributive law]
=(A'+ 0).(A'+ B + C'.D) [Complement law]
=A'(A' + B + C'.D) [Properties of 0 and 1]
=A'.A' + A'.B + A'.C'.D [Distributive law]
=A' + A'.B + A'.C'.D [Idempotent law]
4646 Touchpad Computer Science-XII

