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4. 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
The final column results in 1. So, it is a tautology. Hence proved.
5. If, A = “The Taj Mahal is one of the seven wonders of the world” and
B = “It is a favourite tourist destination”
Then, write the statements for the following propositions:
(i) A ∧ B (ii) A → B (iii) A ↔ B (iv) A' ∨ B' (v) B' → A
Ans. i. The 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. The Taj Mahal is one of the seven wonders of the world if and only if it is a favourite tourist destination.
iv. The 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.
6. Draw the truth table of (A ⊙ B) ⊕ C.
Ans. A B A ⊙ B C (A ⊙ B) ⊕ C
0 0 1 0 1
0 0 1 1 0
0 1 0 0 0
0 1 0 1 1
1 0 0 0 0
1 0 0 1 1
1 1 1 0 1
1 1 1 1 0
7. Prove using truth table (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C).
Ans. A B A ⊕ B C (A ⊕ B) ⊕ C B ⊕ C A ⊕ (B ⊕ C)
0 0 0 0 0 0 0
0 0 0 1 1 1 1
0 1 1 0 1 1 1
0 1 1 1 0 0 0
1 0 1 0 1 0 1
1 0 1 1 0 1 0
1 1 0 0 0 1 0
1 1 0 1 1 0 1
We find that the two columns are identical. Hence proved.
Unsolved Questions
A. Tick ( ) the correct option:
1. The propositional operator → represents ………………… .
a. conjunction b. implication
c. disjunction d. negation
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Propositional Logic, Hardware Implementation, Arithmetic Operations 91

